Vertexformer: the interpretable predictive research on thermocapillary convection of large Prandtl number liquid bridges

Abstract

This study investigates thermocapillary convection in large-Prandtl-number liquid bridges under microgravity. A novel, interpretable deep learning framework, Vortexformer-SHAP, is developed to accurately predict flow dynamics and uncover underlying physics from experimental data. The lightweight Vortexformer model, with 0.02 billion parameters, employs spatiotemporal attention mechanisms and achieves superior performance (R² > 0.99, RMSE = 0.2213) compared to baseline models. SHAP-based interpretability analysis rediscovers the Geometry Effect and reveals two key findings. First, the bifurcation effect of cold-end bridge temperature suggests the critical role of solid heat transfer relevant to the floating-zone liquid bridge. Second, an intrinsic oscillatory thermocapillary convection periodic law of 10–15 time steps (approximately 5–7 s) is identified within the liquid bridge geometric range of the experiment. This work provides a new paradigm integrating a customized deep learning model with physical interpretation, offering both a technical foundation and theoretical insights for real-time prediction and intelligent control of complex thermal-fluid systems in space.

Introduction

Forces acting on fluids can be categorized into body forces and surface forces. At macroscopic scales under standard Earth gravity, body forces typically dominate. However, within aerospace systems, particularly during manned spaceflight, where operations occur in a microgravity environment, gravitational effects such as buoyancy-driven convection, sedimentation, and pressure gradients induced by Earth's gravity are significantly diminished. Under these conditions, surface forces become predominant in determining fluid behavior. For living and working in manned space structures (e.g., space shuttles, space stations, and potential future space colonies), a deeper understanding of fluid behavior under microgravity, specifically surface force-dominated flows, is essential, as the system dynamics differ significantly from those on Earth.

The liquid bridge serves as a classic and ideal model for studying surface force-dominated flows under microgravity. It refers to the liquid column formed and sustained between two solid surfaces, primarily by surface tension forces. Naturally occurring liquid bridges are widespread, such as fluids trapped within porous media, and their dynamics are crucial for industrial applications, including petroleum recovery^1–5 and multiphase catalysis⁶.

In space, stable gas/liquid free surfaces or liquid/liquid interfaces exist in a state of equilibrium maintained by interfacial tension. Surface tension depends on factors such as temperature, solute concentration, and electric potential. Gradients in surface tension result from gradients in these physical quantities. Therefore, in a quiescent, non-externally-forced interfacial system in space, it is typically the inhomogeneity of physical quantities that induces surface forces, ultimately driving fluid motion. This phenomenon is known as the Marangoni effect. More specifically, when a temperature gradient is applied, the gas-liquid free surface of a liquid bridge is subjected to a non-equilibrium Marangoni effect. This temperature-driven flow along the free surface is termed thermocapillary convection, which is the core focus of this study⁷.

Research on thermocapillary convection holds significance not only for fundamental physics but is also directly relevant to space-based floating-zone crystal growth. The floating-zone technique is an advanced zone-melting method for crystal purification and has been primarily used since the early 1950s for growing high-purity, high-quality semiconductor crystals⁸. During the single crystal growth process via the floating-zone method, a molten zone forms a liquid bridge between the polycrystalline feed rod and the newly formed single crystal. As the heating zone (the float zone) moves, the melt solidifies directionally, yielding a single crystal. A major advantage of this technique is its “containerless” nature, which completely eliminates the issue of melt contamination by crucible wall materials that can degrade crystal purity in traditional crucible-based methods. However, on Earth, gravity imposes stringent constraints on the liquid bridge; as its size increases, its own weight can readily overcome the confinement provided by surface tension, leading to rupture. This makes the fabrication of large single crystals extremely difficult. In microgravity, this fundamental limitation is removed, rendering the floating-zone technique one of the most promising methods for space materials processing.

In Pearson's seminal work⁹, they pioneered the theoretical analysis of instability within a liquid layer bounded by a free surface and a heated solid boundary under microgravity conditions, demonstrating that convection occurs when the temperature difference exceeds a critical value. In 1978, Schwabe et al.¹⁰ and Chun and Wuest¹¹ experimentally observed that when the temperature difference across a liquid bridge surpasses a critical threshold, the initially stable laminar thermocapillary flow loses stability and transitions into complex three-dimensional oscillatory flow. This oscillatory flow has been confirmed as the root cause of inhomogeneities within crystals, as it induces periodic melting and solidification at the solid-liquid interface, thereby affecting the uniform transport of solutes. During the D-1 and D-2 space missions and sounding rocket experiments, researchers conducted thorough studies on thermocapillary flow during floating-zone crystal growth^12–14, proving that striation defects in crystals grown under microgravity are directly caused by oscillatory thermocapillary flow. Therefore, deeply understanding and ultimately controlling this oscillatory flow is of crucial strategic importance for improving the quality of space-grown crystals and developing advanced space materials processing technologies.

To simplify research, the half-floating-zone liquid bridge model is widely adopted in academia. This model uses two coaxial solid disks maintained at different constant temperatures to confine the liquid bridge. In experimental studies using half-floating-zone liquid bridges, real semiconductor melts (such as molten silicon) are challenging for precise optical measurement and flow visualization due to their high melting point, susceptibility to surface oxidation, and opacity. Consequently, researchers typically employ transparent model fluids with good optical properties, such as silicone oil or molten salts. However, these model fluids differ significantly from semiconductor melts in a key dimensionless parameter, the Prandtl number (Pr). The Prandtl number is defined as:

$$\mathrm{Pr}=\frac{\nu }{\alpha }$$ 1

where $\nu$ is the kinematic viscosity and $\alpha$ is the thermal diffusivity. It represents the ratio of momentum diffusivity to thermal diffusivity, which significantly influences the thermocapillary oscillation mechanism. Semiconductor melts are typical low-Prandtl-number fluids (Pr ≪ 1), with both $\nu$ and $\alpha$ being relatively large, allowing heat to diffuse rapidly and making the flow inertia-dominated. In contrast, silicone oils are high-Prandtl-number fluids (Pr ≫ 1), where heat diffusion is slow, and heat is primarily “frozen” within the fluid and transported by convection. This leads to strong coupling between the momentum and temperature fields. The instability in such fluids is believed to be initiated by a traveling wave known as a “hydrothermal wave,” characterized by the critical Marangoni number:

$$\mathit{Ma}=\frac{{\sigma }_T^{\prime }\mathrm{\Delta }T\mathrm{H}}{\mu \alpha }$$ 2

where ${\sigma }_T^{\prime }$ is the temperature coefficient of interfacial tension, $\mathrm{\Delta }T$ is the temperature difference, $H$ is the characteristic length, and $\mu$ is the dynamic viscosity.

The classical hydrothermal wave instability theory was first proposed by Smith and Davis¹⁵ through linear stability analysis of an infinite horizontal liquid layer and was later applied by Xu and Davis¹⁶ to an infinite-length liquid bridge model. The best way to validate these theories based on idealized infinite models is through experiments with long liquid bridges in space. Schwabe, in the MAXUS-4 rocket experiment¹⁷, investigated a large aspect ratio (Ar = 2.5) liquid bridge near the Rayleigh stability limit, successfully observed hydrothermal waves, and found that the critical conditions for oscillation were in excellent agreement with the theoretical predictions of Xu and Davis¹⁶. This success validated the theory under idealized conditions.

However, when the geometry of the liquid bridge deviates from the idealized infinite cylinder, experimental results show that the actual critical Ma number (Ma_c) can be orders of magnitude larger than theoretical predictions¹⁶. Studies indicate that key geometric parameters, the Aspect Ratio (Ar) and Volume Ratio (Vr), significantly influence the Ma_c. Vr and Ar are defined as follows:

$$\mathit{Vr}=\frac{V}{V_0}$$ 3

where $V_0$ is the volume of the gap between the two cylinders and V is the volume of the working fluid.

$$\mathit{Ar}=\frac{\mathrm{H}}{\mathrm{D}}$$ 4

where $H$ is the height of the liquid bridge and $D$ is the diameter of the supporting disks.

Regarding the Ar, early studies reached completely contradictory conclusions, suggesting that the critical Marangoni number either monotonically decreases¹⁸ or monotonically increases¹⁹ with increasing Ar. Although recent research has revealed more complex non-monotonic trends^20,21, the newly discovered phenomenon of “jumps” in critical conditions²² further highlights our insufficient understanding of the mode competition mechanism. Similarly, concerning the influence of the Vr, a long-standing debate exists in academia over the existence of a stable “gap”, forming opposing viewpoints of “gap existence”^23,24 versus “no gap, but continuous variation”²⁵. Subsequent studies suggest this issue may be related to the Prandtl number and the gravity environment itself^26–28.

It is generally accepted that for large-Pr fluids, the analysis based on hydrothermal instability theory holds. As a contrasting viewpoint, hydrodynamic instability theory has been proposed to explain the behavior of low-Pr fluids, suggesting that their instability is driven by a two-step bifurcation process initiated by flow inertial forces²⁹, which is fundamentally different from the hydrothermal wave mechanism. Furthermore, numerical studies indicate that the two-step transition process caused by hydrodynamic instability may also apply to large-Pr fluids^30,31. In practical space-based floating-zone crystal growth, it is necessary to combine these two theories, which means applying the well-developed understanding from large-Pr fluid experiments to control low-Pr fluids. Additionally, theories based on free surface deformation causing flow instability exist^32,33. However, the extremely small free interface deformations observed in experiments^33–37 have cast doubt on this mechanism being the primary cause of flow instability.

Kang et al. conducted a batch experiment of space-based half-floating-zone liquid bridge aboard the Chinese Space Station (CSS) Tiangong-2 (TG-2) module, ultimately completing 740 sets of experimental runs. Analysis of the data revealed the systematic influence of parameters such as aspect ratio and volume ratio on the instability of thermocapillary convection, indicating the absence of a stable “gap” in large-Pr liquid bridges, which partially resolves the aforementioned debate^38,39. Furthermore, the study summarized the diverse transition pathways of space-based thermocapillary convection⁴⁰. This high-quality, large-scale experimental dataset, combined with prior theoretical analyses, constitutes a solid foundation for deep learning research.

Deep learning models have demonstrated exceptional predictive and control capabilities in engineering fields⁴¹. And establishing human-operated production environments in space is extremely difficult, whereas unmanned factories present significantly lower complexity and cost. This makes constructing deep learning models particularly suitable for production control under space conditions. On the other hand, data-driven science, as the fourth paradigm of scientific research, has enabled numerous scholars in fundamental physics to uncover hidden physical properties that are difficult to discover with traditional methods⁴².

The evolution of neural network architectures has been pivotal to these advances. Early feedforward networks and convolutional neural networks (CNNs)⁴³ excel at spatial feature extraction but struggle with long-range dependencies in sequential data. Recurrent neural networks (RNNs)⁴⁴ and their gated variants (LSTM and GRU)^45,46 were developed to model temporal dynamics, yet they suffer from gradient vanishing and limited parallelization. A transformative breakthrough came with the introduction of the Transformer architecture⁴⁷, which replaces recurrent operations with a multi-head self-attention mechanism. This mechanism computes attention weights by mapping input sequences into queries, keys, and values, allowing the model to dynamically weigh the importance of different positions regardless of their distance. The result is both improved capture of long-range dependencies and significantly enhanced training efficiency through parallel processing. The attention mechanism has since become a cornerstone of modern deep learning.

To fully explore the potential within the high-precision space experiment data, this study modifies the general Transformer architecture⁴⁷ to incorporate spatiotemporal characteristics, designing and training a deep model named Vortexformer.

Based on this model, the core tasks of this research are twofold: first, to accurately model and predict the temperature oscillation signals in half-floating-zone liquid bridge thermocapillary convection. This provides the foundational model for control, marking the first step towards the regulation of the liquid bridge in space. Second, building upon accurate prediction, to advance the interpretability analysis of the model, aiming to identify underlying physical mechanisms from its high-dimensional, complex parameters with SHAP (SHapley Additive exPlanations)^48,49. This is intended to provide novel theoretical insights into the physical principles of thermocapillary convection and the optimized control of space complex thermal-fluid systems. In summary, the fundamental goal of this study is to transcend the traditional research paradigms of phenomenological conclusions based on control parameter interactions and descriptive transition pathways by constructing a deep model to achieve both prediction and a more fundamental analysis of the thermocapillary convection behavior in half-floating-zone liquid bridges.

Methods

A Vortexformer-SHAP research framework integrating data-driven approaches and physical interpretation is employed to uncover the spatial correlations within the temperature oscillations. The implementation of the model is detailed in the following three sections: the “Vortexformer model development”, the SHAP method, and the “Specialized Vortexformer for interpretability”.

Physical experiments

The space experimental setup was introduced by Kang et al.³⁸, as illustrated in Fig. 1. Both supporting rods are made of copper, with a diameter of D = 20 mm. One rod is heated to a temperature T_up via an electric heating film, while the other is maintained at a temperature T_down by a thermoelectric cooler. When a temperature difference ΔT is applied, convection is observed within the liquid zone. Once ΔT exceeds a certain critical threshold (ΔT_C), the flow becomes unstable and transitions into an oscillatory state. Experiments indicate that the entire transition can progress through various pathways, evolving from steady state, periodic oscillation, and quasi-periodic states, to ultimately chaotic flow. Different heating rates were set for different experiments, starting from an initial ΔT of 0 °C. Once a predetermined ΔT is reached, the system maintains the respective high and low-temperature settings, as shown in Fig. 1 (Right)⁴⁰.

Fig. 1. Schematic diagram of the surface morphology and measurement points distribution of the liquid bridge in the space experiment.

The liquid bridge and its surface morphology in the space experiment (Left)³⁸. Reproduced with permission from Kang et al.38. Copyright 2019 Cambridge University Press. The schematic diagram of the liquid bridge and temperature measurement points (Right)⁴⁰. Reproduced with permission from Kang et al.⁴⁰. Copyright 2022 American Physical Society.

Five thermocouples are used to measure temperatures within the liquid bridge, while the temperatures of the cold and hot rods are measured by two additional thermocouples. The thermocouples used are type K with a diameter of 0.02 mm. The five internal thermocouples are positioned at consecutive angles of 0°, 90°, 180°, 270°, and 315°, and are labeled T₁– T₅, as shown in Fig. 1 (Right)⁴⁰.

The KF96L 5cSt silicone oil with Pr = 67.29; Density ρ = 915 kg/m³; Thermal diffusivity $\alpha$ = 0.12 × 10⁻⁶ m²/s; Kinematic Viscosity ν = 5.0 × 10⁻⁶ m²/s; Temperature Coefficient of Surface Tension ${\sigma }_T^{\prime }$ = 6.58 × 10⁻⁵ N·m⁻¹·K⁻¹, is selected as the working fluid. The silicone oil is stored in a cylinder, and a piston inside the cylinder is used to inject or withdraw the oil to form liquid bridges of different volumes. The piston is driven by a motor from Power Integrations (PI), allowing precise control of the liquid volume V. Simultaneously, the distance H between the two rods is adjustable, enabling the acquisition of more diverse parameter sets.

The geometry of the liquid bridge can be uniquely determined by the dimensionless volume ratio Vr and the height H. The range for Vr is 0.54–1.15, and H varies from 4 mm to 22 mm. After pre-setting the target Vr, H, heating rate, and target temperatures on the ground, the CSS TG-2 module liquid bridge experimental system executed the runs automatically, synchronously initiating temperature acquisition at a sampling frequency of 2 Hz. Experimental results can utilize either the resistor heating process or the natural cooling process. The natural cooling process is generally considered to involve less external interference and is closer to a natural process. Although phase differences exist among the signals from different thermocouples, their temperature oscillation characteristics exhibit consistency. Therefore, for the single-point analysis in this paper, data from thermocouple T₁ is uniformly selected for in-depth discussion. An example dataset from one experiment, including both heating and natural cooling phases, is presented in Fig. 2 for a case with Vr = 0.76 and H = 18 mm.

Fig. 2. Temperature profiles over time.

Temperature signals vary with time, exemplified for a case with Vr = 0.76 and H = 18 mm during the heating and natural cooling phases.

Data preparation

The dataset is established based on the experimental data of thermocapillary convection in liquid bridges from the TG-2 experiment, as introduced in the “Physical experiments” section. All selected data are shown in Table 1:

Table 1.

Features examples. The columns represent Features (F), and the leftmost column is time (t)

F t	T1–5 °C	Tup °C	Tdown °C	H mm	Vr
1	27.7	54.4	21.7	4	0.7
2	27.7	54.4	21.7	4	0.7
……	……	……	……	……	……
N	13.2	18.4	17.2	4	0.7
N + 1	13.2	18.4	17.2	4	0.7

The experimental data is a complete and continuous time series. To ensure the model effectively learns the spatiotemporal evolution characteristics within thermocapillary convection, the dataset partitioning preserves the integrity of the experimental sequences. Specifically, each experimental run, defined by a unique combination of H and Vr, is assigned an independent identification number. A total of 186 valid experimental runs were conducted, corresponding to 156 unique H and Vr combinations, as some conditions were repeated. These 156 combinations cover the H and Vr parameter space of the experiment, although the intervals between adjacent values are not uniform. The dataset is then split according to the proportion of these experimental IDs. 70% of the experimental sequences are used as the training set, while the remaining 30% are equally divided into the test set and the validation set, each accounting for 15%. This approach ensures that the model encounters complete, uninterrupted experimental processes during both training and evaluation.

Regarding data preprocessing, since the experimental data are complete, continuous, and free of outliers, Input Normalization was only applied to all input features. This preprocessing step standardizes the input data to a consistent scale and distribution, which facilitates faster and more stable convergence during model training and provides a stable foundation for learning. The Input Normalization operation can be expressed mathematically, for a total of input x, its normalization process is:

$$\hat{x}=\frac{x-\mathrm{\mu }}{\sqrt{{\mathrm{\sigma }}^2+\mathrm{ϵ}}}$$ 5

where $\mu$ and $\mathrm{\sigma }$ are the mean and standard deviation of the total data, respectively; and $ϵ$ is a small constant added for numerical stability.

Spatial-temporal attention

The conventional attention mechanism in transformer architectures operates through the interaction between Query (Q) and Key (K) matrices to compute correlation scores between sequence elements in a sequence. Q and K are similar to the search content and results in a search engine. The standard attention operation can be formalized as:

$$\text{Attention}\left(Q,K,V\right)=\mathrm{softmax}\left(\frac{Q{K}^T}{\sqrt{d_k}}\right)V$$ 6

where $Q\in R^{n\times d_k}$, $K\in R^{m\times d_k}$, and $V\in R^{m\times d_v}$ represent the query, key, and value matrices, respectively, with $d_k$ denoting the dimension of the key vectors and $n$, $m$ representing sequence lengths.

Specifically, first, the dot product $Q{K}^{\mathrm{\top }}$ produces a similarity matrix of size $n\times m$, where each entry measures the compatibility between a query and a key. Second, this matrix is scaled by $1/\sqrt{d_k}$ to prevent the dot products from growing large in magnitude. Third, the softmax function is applied row-wise to convert the similarity scores into a probability distribution over the key positions. Finally, these attention weights are used to compute a weighted sum of the value vectors, producing an output that focuses on the most relevant parts of the input. In this formulation, the Query vector acts as a representation that seeks relevant contextual information, while the Key vector serves as an identifier that matches against the Query to determine relevance. The Value vector carries the actual information to be propagated. This design enables the model to dynamically assign importance to different positions in the sequence, regardless of their distance⁴⁷.

As stated in the “Physical experiments” section, T_1–5 has a spatial distribution within the liquid bridge. To capture the spatial correlations inherent in thermocapillary convection patterns, we introduce a Spatial Attention (SA) mechanism. At each time instant t, the temperature readings from all five measurement points are simultaneously encoded into both the Q and K components:

$$Q_{\text{spatial}}^{\left(t\right)}=W_Q\left[T_1^{\left(t\right)},T_2^{\left(t\right)},T_3^{\left(t\right)},T_4^{\left(t\right)},T_5^{\left(t\right)}\right]$$ 7

$$K_{\text{spatial}}^{\left(t\right)}=W_K\left[T_1^{\left(t\right)},T_2^{\left(t\right)},T_3^{\left(t\right)},T_4^{\left(t\right)},T_5^{\left(t\right)}\right]$$ 8

The spatial attention weights are then computed as:

$${\mathrm{\alpha }}_{\mathit{ij}}^{\text{spatial}}=\frac{\exp \left(Q_i^{\text{spatial}}\cdot K_j^{\text{spatial}}/\sqrt{d_k}\right)}{{\sum }_{j=1}^5\exp \left(Q_i^{\text{spatial}}\cdot K_j^{\text{spatial}}/\sqrt{d_k}\right)}$$ 9

where ${\mathrm{\alpha }}_{\mathit{ij}}^{\text{spatial}}$ quantifies the correlation between measurement points $i$ and $j$ at the same time instant. This formulation corresponds to the Softmax function part in Eq. (6), enabling the model to learn how temperature variations at one spatial location influence or correlate with temperatures at other locations simultaneously, thereby capturing the spatial structure of the convection patterns.

To model temporal dependencies, we employ a Time-to-Time Attention (TTA) module, analogous to self-attention in standard transformers⁴⁷. For a sequence of features, consecutive data points $x_i,x_{i+1},\dots ,x_t$ are processed simultaneously:

$$Q_{\text{temporal}}=W_Q^{\text{temp}}{X}_{\text{seq}},K_{\text{temporal}}=W_K^{\text{temp}}{X}_{\text{seq}},V_{\text{temporal}}=W_V^{\text{temp}}{X}_{\text{seq}}$$ 10

where $X\in R^{S\times d}$ represents the input sequence matrix. The temporal attention is computed as:

$$\text{TTA}\left(X_{\text{seq}}\right)=\text{Softmax}\left(\frac{Q_{\mathit{temporal}}{K}_{\mathit{temporal}}^T}{\sqrt{d_k}}\right)V_{\text{temporal}}$$ 11

This formulation enables the model to learn correlations between different time steps within each sequence, capturing temporal dependencies and evolution patterns in the thermocapillary flow dynamics.

The above is the method of achieving Spatial-temporal Attention through SA and TTA from a mathematical perspective.

Vortexformer model development

This section will elaborate on other specific corrections and implementations by introducing the Vortexformer global architecture. The complete global architecture is presented in Fig. 3, and the training details are placed at the end of the section.

Fig. 3. Global architecture of Vortexformer.

The model is mainly composed of an encoder and a decoder. Within these modules, different types of attention mechanisms are employed to implement distinct functions.

Prior to being fed into the model, all input features are embedded into a 256-dimensional space (D). A specialized invariant embedding scheme is employed for the temperature readings from thermocouples T₁ to T₅, designed to respect the rotational symmetry of the liquid bridge system. This scheme incorporates the known angular positions of the thermocouples $\theta \in \{0°,90°,180°,270°,315°\}$. The invariant embedding is formulated as:

$$D_{\mathit{inv}}\left(\mathrm{\theta }\right)=2\sin \left(\frac{\mathrm{\theta }}{4}\right)-1$$ 12

The final embedding for a temperature feature is the concatenation of a standard 255-dimensional embedding with this 1D invariant embedding:

$$D_T=\mathit{Concat}\left(D_{255},D_{\mathit{inv}}\right)$$ 13

Thus, the input embedding stage processes features through two parallel paths: the standard embedding for most features and the specialized invariant embedding for the T₁–T₅ temperatures. This design provides the model with an explicit representation of the thermocouples' circumferential arrangement, which was absent from the raw scalar inputs, effectively establishing a spatial coordinate system within the feature space. The fixed angular encoding serves as a physical constraint, guiding the model to leverage multi-sensor phase correlations when reconstructing the spatiotemporal dynamics of the rotating hydrothermal wave. It reduces the burden to implicitly discover this geometric relationship from data alone, lowers the risk of converging to a physically implausible local optimum during training, and improves learning efficiency.

Following embedding and positional encoding, the features are routed differently: standard features proceed directly to the Feature Aggregate (FA) module, while the embedded T₁–T₅ temperatures are first processed by the Spatial Attention (SA) module. The model employs two blocks of SA, each equipped with exceptionally wide feedforward neural networks. These feedforward networks operate at ten times the base embedding dimension, resulting in intermediate layer dimensions of $D_{\text{FFN}}^{\text{SA}}=10D=2560$ neurons. The expanded capacity of these wide architectures enables the spatial attention modules to capture rich and complex spatial correlations across measurement points, accommodating the intricate spatial dependencies present in convection patterns.

Subsequently, all features are unified in the FA module, implemented as a single multi-head attention layer followed by a feedforward network with an intermediate dimension of $4D=1024$ neurons.

Then all the features are input to the TTA module. The TTA module processes multiple physical quantities simultaneously through parallel feature channels. For $F$ distinct features, each with its own temporal sequence, we generate a stack of TTA matrices as ${\text{TTA}}_1,{\text{TTA}}_2,\dots ,{\text{TTA}}_F$, where each ${\text{TTA}}_f$ corresponds to the temporal attention patterns of feature $f$. This stacked matrix structure allows the model to process multiple physical quantities in parallel, with each feature sequence producing attention patterns that capture the time-wise correlations specific to that variable, enabling a comprehensive representation of the multi-physics nature of thermocapillary convection.

A total of eight blocks of TTA were deployed, arranged in a deep sequential structure. These TTA employ comparatively narrower feedforward networks operating at four times the base D, yielding intermediate layer dimensions of $D_{\text{FFN}}^{\text{TTA}}=4D=1024$ neurons per layer. This deep, narrow design enables the model to progressively refine temporal patterns across multiple layers.

The above is the encoder part. The decoder is set to have the same scale as the encoder, which is used to achieve good performance in sequence-to-sequence prediction⁵⁰. The sequence-to-sequence prediction is one of the techniques of autoregressive, which predicts the next element based on the previously generated output, until the entire sequence is completed.

All the parts in the model use multi-head attention with 8 heads. Standard transformer components, such as layer normalization and residual connections, are retained⁴⁷. The sequence length is set to 12, a value identified as optimal by an exhaustive search over possible lengths using a simple LSTM model, as detailed in the “Benchmark test” section and visualized in Fig. 19. Specific implementations of linear projection steps are not elaborated here.

Fig. 19. Trained on an LSTM model.

Best sequence length between 10 and 15 time steps.

During the training phase, the model is optimized using the MSE (mean square error) loss function computed between the ground truth and predicted values from all five temperature measurement points. The models were trained on one machine with an NVIDIA A800 GPU. The optimization is performed using the AdamW optimizer and the early stopping mechanism. Eventually, the training was stopped after approximately 1500 steps, and the total training duration was 72 h. Then, a Vortexformer was derived, with the total parameter count constrained to 0.02B. For inference, the trained model supports multiple operational modes: single-step prediction at individual measurement points, autoregressive forecasting over extended periods, and simultaneous multi-point prediction, providing flexibility for different application scenarios.

SHAP method

The mathematical principle of the SHAP method for quantifying the marginal contribution of each measurement point temperature to the predicted value is based on the Shapley value from game theory^51,48. A straightforward explanation of the SHAP method is that it distributes the total effect among all independent input features based on their respective related weight parameters in the model. The calculation formula for the Shapley value ${\varphi }_i$ is as follows⁴⁹:

$${\varphi }_i={\sum }_{S\subseteq F\setminus \left\{i\right\}}\frac{\left|S\right|!\left(N-\left|S\right|-1\right)!}{N!}\left[f\left(S\cup \left\{i\right\}\right)-f\left(S\right)\right]$$ 14

where, $F$ is the set of all features, with a total of $N$ features; $S$ is any subset $S\subseteq \mathit{F\backslash }\left\{i\right\}$ that does not contain feature $i$; $\mathrm{\mid }S\mathrm{\mid }$ is the size (number of features) of subset $S$; $f\left(S\right)$ is the model predicted value using only the features in subset $S$; $f(S\cup \{i\left\}\right)$ is the model predicted value after adding feature $i$.

The formula can be understood in two parts. The term $f(S\cup \{i\left\}\right)-f\left(S\right)$ represents the marginal contribution, indicating the incremental contribution of feature $i$ to the model prediction upon its addition to subset $S$. The coefficient $\frac{\mathrm{\mid }S\mathrm{\mid }!(N-\mathrm{\mid }S\mathrm{\mid }-1)!}{N!}$ is the weighting factor, used to average over all possible orders in which features can be added. The numerator counts the number of permutations where feature $i$ is added after the features in subset $S$. The denominator is the total number of all possible feature permutations $N!$.

Finally, summing all ${\varphi }_i$ linearly yields the model prediction $f\left(x\right)$⁴⁸$:$

$$f\left(x\right)={\varphi }_0+{\sum }_{i=1}^N{\varphi }_i$$ 15

where ${\varphi }_0$ is the featureless reference value or Base Value, which is the model output when no features are input. Through the above calculation, it is evident that the predicted value can be decomposed into contributions from each input feature.

Specialized Vortexformer for interpretability

To construct an interpretable model, as titled “Arena for Parameters”, we expanded the original feature set by deriving additional parameters from the existing features. All possible transformations and combinations that could be conceived were included without a predefined selection criterion, with the aim of maximizing feature diversity. Keeping all other experimental settings unchanged, we retrained the model using this augmented feature space. The complete list of features is provided in Table 2, and the definitions of the newly introduced features are also given in Table 2. This expansion was intended to supply the SHAP analysis with a richer set of variables for evaluating feature contributions. Although this approach introduces data leakage, it was deemed unavoidable in order to establish meaningful connections between the predicted temperature values and well-understood physical quantities.

Table 2.

Adding Features and Feature Definitions

Adding features	Definition
Prandtl number (Pr)	Pr = μ / (ρ × κ)
Marangoni number (Mat)	Mat = ${\sigma }_T^{\prime }$ΔTH / μα
Dimensionless heating rate (Qrate)	Qrate = (dTup / dt) × H2 / (ΔTmax × κ)
Control temperature gradient (dT / dt)	dT / dt [Tup (t) − Tup (t − 1)] / Δt
Rolling standard deviation (Tstd_roll)	Tstd_roll(t) = σroll (TX [τ])tτ=t−n+1, n = 2048
Lagged temperature (TXlag_n)	TXlag_n(t) = TX(t − k), k = 1, 5, 10, X = 1,5
Mean temperature value (Tmean)	Tmean(t) = mean(T1(t), T2(t), …, T5(t))
Spatial variance (Tvar)	Tvar(t) = variance(T1(t), T2(t), …, T5(t))
Vertical gradient (Tgrad)	Tgrad = (Tup − Tdown) / H
PSD (power spectral density) feature	Domain frequency & power
Spectral entropy (Sentropy)	Shannon entropy of the normalized Power Spectral
Local Lyapunov	Lyapunov exponent within 2048 time steps
Global Lyapunov	The Lyapunov exponent of a certain condition

All the features underwent an ergodic SHAP analysis. However, during the analysis in the “Rediscover geometry effect” and “Enlightening insights” sections, only a subset of these derived features yielded meaningful insights. The majority did not contribute substantively to model interpretability. It should be noted that only the features defined in this section and in the “Physical experiments” section are directly considered in the analysis. The influence of other physical effects, such as heat transfer through the liquid bridge surface, the azimuthal mode structure, and so on, is ultimately attributed to the configured features.

Metrics

To rigorously evaluate the predictive accuracy and reliability of the Vortexformer model, we employ two complementary statistical metrics that quantify different aspects of model performance: the coefficient of determination (R²) and the root mean square error (RMSE).

The coefficient of determination, denoted as R², measures the proportion of variance in the observed data that is successfully explained by the model predictions. This metric provides a normalized assessment of model performance that remains interpretable across different physical quantities and measurement scales. The R² score is computed as:

$$R^2=1-\frac{{\sum }_{i=1}^N{\left(y_i-\widehat{y_i}\right)}^2}{{\sum }_{i=1}^N{\left(y_i-\stackrel{\circledR }{y}\right)}^2}=1-\frac{{\mathit{SS}}_{\mathit{res}}}{{\mathit{SS}}_{\mathit{tot}}}$$ 16

where $y_i$ represents the $i$-th observed value, $\widehat{y_i}$ denotes the corresponding model prediction, $\stackrel{\circledR }{y}=\frac{1}{N}{\sum }_{i=1}^N{y}_i$ is the mean of observed values, $N$ is the total number of samples, ${\text{SS}}_{\text{res}}$ represents the residual sum of squares, and ${\text{SS}}_{\text{tot}}$ denotes the total sum of squares.

The R² metric ranges from negative infinity to unity, where a value of one indicates perfect prediction accuracy, zero suggests that the model performs no better than a naive baseline that predicts the mean value for all samples, and negative values indicate performance worse than this baseline. In the context of thermocapillary convection prediction, R² values approaching unity demonstrate that the model successfully captures the underlying dynamics and variability patterns in temperature evolution and flow characteristics. This metric is particularly valuable when comparing model performance across different features with disparate scales and units, as it provides a dimensionless measure of predictive quality.

The RMSE quantifies the absolute magnitude of prediction errors in the original units of measurement, providing direct insight into the practical accuracy of model predictions. The RMSE is defined as:

$$\mathit{RMSE}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(y_i-\widehat{y_i}\right)}^2}$$ 17

This metric represents the standard deviation of prediction residuals and maintains the same dimensional units as the predicted quantity. For temperature predictions measured in degrees Celsius, the RMSE directly indicates the typical magnitude of prediction error in thermal units. Lower RMSE values correspond to more accurate predictions, with the metric being particularly sensitive to large errors due to the quadratic penalty applied to individual residuals.

The RMSE serves as a critical complement to the R² metric by providing absolute error magnitudes. While R² indicates how well the model captures variance patterns, RMSE directly quantifies whether the prediction errors fall within acceptable bounds. Together, these two metrics offer a comprehensive evaluation framework.

Results

Benchmark Test

This section conducts predictive analysis by Benchmark Test of the metrics. To systematically evaluate the predictive performance of Vortexformer, this study established two comparative models.

First, the Long Short-Term Memory (LSTM) model, known for its strong performance in time-series forecasting, was selected as a representative of classical deep models. Before inputting into the LSTM, four preprocessing techniques, including low-pass filtering, Kalman filtering, wavelet denoising, and Savitzky–Golay filtering, were applied to the raw data. The model demonstrating the best performance among those trained on the raw data and filtered data was chosen to represent the final performance of LSTM. Second, to further examine the effectiveness of Vortexformer after incorporating modeling for spatiotemporal flow characteristics, a standard Transformer model with a comparable parameter count was selected for comparison. In this model, all feature sequences were fed directly into multi-head attention modules. The training process of all models is consistent with that described in the “Vortexformer model development” section.

Under non-autoregressive prediction settings, the performance of the three models in terms of R² and RMSE is shown in Table 3. As can be seen from the table, Vortexformer significantly outperforms the other two models in both R² and RMSE metrics, achieving an R² of 0.9915 and an RMSE of 0.2236, demonstrating extremely low prediction error. Both the standard Transformer and LSTM models attained R² values of only around 0.44–0.46, as indicated in Eq. (16). This suggests that their Residual Sum of Squares accounts for more than half of the Total Sum of Squares.

Table 3.

Performance metrics of the three models on the test set

MetricsModels	R²	RMSE
Vortexformer	0.9915	0.2236
Transformer	0.4410	0.8671
LSTM	0.4600	1.2813

In the single-step prediction task, as the RMSE values of all three models are relatively low, the direct comparison between true and predicted values is less apparent. Therefore, the residual distributions are presented in Fig. 4. The definition is Residual = True − Predicted. It is evident that the Vortexformer residual distribution is narrower and more concentrated, indicating that this model possesses a more precise ability to capture key dynamic features such as oscillation period, amplitude, and phase. Further confirming in Fig. 5, the true and predicted values from Vortexformer almost completely overlap.

Fig. 4. Residual distribution of the three models on the test set.

(A) represents the Vortexformer, (B) represents the Transformer, and (C) represents the LSTM.

Fig. 5.

Prediction on test set. Comparison of the 700,000-step prediction with the true values on the test set.

To comprehensively evaluate the effectiveness of the improvements in Vortexformer spatiotemporal attention, a comparative analysis was conducted between it and the standard Transformer with equivalent parameter counts under autoregressive prediction mode. It should be noted that the LSTM model employed in this study was not architecturally designed for autoregressive prediction. Therefore, this comparison is conducted solely between Vortexformer and the standard Transformer.

When autoregressive prediction is employed for continuous forecasting, the left and right panels of Fig. 6 illustrate the stepwise prediction errors of the Vortexformer and the standard Transformer, respectively. The Fig. 7 displays an overall error heat map over multiple consecutive prediction steps. It can be observed from the two kinds of error graphs that the maximum error value of Vortexformer lies within 1.25, with a stable distribution. In contrast, the maximum error of the Transformer reaches 4, and errors occur in both positive and negative directions. As shown in Fig. 8, in terms of the R², Vortexformer maintains a stable value close to 1 throughout the prediction zone, indicating consistently high accuracy. In contrast, the Transformer shows a sharp decline in performance, with its R² dropping abruptly from 0.4 to −4.

Fig. 6. Box plots of errors for autoregressive predictions on the test set.

Vortexformer (left), Transformer (right).

Fig. 7. Heatmaps of errors for multiple sets of autoregressive predictions on the test set.

Vortexformer (left), Transformer (right).

Fig. 8. Change in R² over prediction time steps on the test set.

Vortexformer (left), Transformer (right).

Physical consistency

To verify the model's physical consistency during the transition process, complete experimental data (Vr = 0.84, H = 12 mm) in the validation set, corresponding to the Ruelle–Takens–Newhouse (RTN) route within the quasi-periodic transition, were selected for prediction. A comparative analysis was conducted on the true-predicted sequences for T₁. The essence of the RTN route lies in the system undergoing at least two Hopf bifurcations, with chaos emerging after the second bifurcation. In the frequency spectrum, this manifests as the emergence of two or more incommensurate frequencies⁵², a phenomenon that has been reported in space experiments of liquid bridges thermocapillary convection⁴⁰.

The top panel of Fig. 9 shows the spectrogram of the experimental data, while the bottom panel shows the spectrogram of the model predictions. It can be observed that before approximately 4000 s, the distribution of frequency components is highly consistent. After 4000 s, the predicted results show a proportionally higher energy content across all high frequencies, then proportionally lower after 7696 s. Additionally, the overall background noise has been suppressed in predictions.

Fig. 9. Spectrogram of temperature siganal.

Spectrogram of True (Top) and Predicted (Down), under specific conditions, Vr = 0.84 and H = 12.

However, this does not affect the identification of fundamental frequency splitting. The three key time points (t = 3791 s, 7696 s, and 9000 s) have been marked, corresponding to the top, middle, and bottom PSD plots in Fig. 10, respectively. These PSD snapshots, obtained using a 2000 s sliding window, clearly illustrate the characteristic features of the RTN transition route: a fundamental frequency f₀ successively splits into f₁ and f₂, which subsequently evolve into three independent frequencies f₁, f₂, and f₃. In Fig. 10, the blue curve represents the true values, and the red curve represents the predicted values. Examining the energy distribution across frequencies, the model predictions maintain good consistency with the experimental data in the low-frequency range. Although a slight energy difference in the high-frequency portion remains observable in the PSD at each moment, the model's accurate reproduction of key dynamical behaviors during the RTN transition process fully demonstrates its good physical consistency.

Fig. 10.

Power spectral density (PSD) of temperature siganal. PSD comparison between experimental measurements (red) and model predictions (blue) at three characteristic time instants: t = 3791 s (top), 7696 s (middle), and 9000 s (bottom), calculated using a sliding window of 2000 samples.

Rediscover the geometry effect

This and the next sections employ a model transition from the Vortexformer to the “Arena for Parameters” with SHAP analysis.

The volume ratio effect reflects the influence mechanism of the dimensionless volume $V_r$ on the system's critical oscillatory behavior. The $V_r$ essentially determines the geometry of the gas-liquid free interface. When $V_r$ is small, the interface is concave, corresponding to a “slender bridge” configuration; when $V_r$ is large, the interface tends to be flat or convex, belonging to a “fat bridge” type. The critical temperature difference ($\mathrm{\Delta }{T}_c$) is defined as the critical parameter that determines the transition from a stable thermocapillary flow to an oscillatory thermocapillary flow. The $\mathrm{\Delta }{T}_c$ under different $V_r$ exhibits a distinct bifurcation characteristic, corresponding to the slender and fat bridge configurations, respectively. In the slender bridge branch, $\mathrm{\Delta }{T}_c$ first decreases and then increases with increasing $V_r$; whereas in the fat bridge branch, $\mathrm{\Delta }{T}_c$ shows a monotonically decreasing trend with increasing $V_r$. A jump change exists between the two branches, manifested as the thermocapillary convection transitioning from low frequency to high frequency^23,38. In related research, the critical Marangoni number ($M{a}_c$) is commonly used to characterize the volume ratio effect. $M{a}_c$ shares the same physical nature as $\mathrm{\Delta }{T}_c$, differing only by a proportional factor related to the liquid bridge height H in this or similar studies.

By conducting a SHAP analysis on data from all experimental conditions through random sampling, an “unexpectedly” volume ratio effect consistent with the aforementioned description was discovered. Figure 11 displays the SHAP dependence plot for the liquid bridge volume ratio $V_r$, where the ranges for $V_r$ and Ma are 0.5 to 0.98 and 0 to 120000, respectively.

Fig. 11. The SHAP value graph of the liquid bridge Vr and Ma.

The yellow dots and lines represent the maximum $\left|{\mathrm{\varphi }}_{\mathrm{i}}\right|$ corresponding to Ma of 120,000.

Stable thermocapillary convection (low Ma) presents as a steady-state flow driven by the surface force of the liquid bridge, being influenced only by temperature difference and the geometric characteristics of the liquid bridge. It is unlikely to have two-branch characteristics. At the transition point and during the fully developed oscillatory thermocapillary convection (high Ma), phenomena such as vortex generation, stretching, and diffusion occur, making the flow characteristics complex, but still with features. In this case, one feature is that a fixed temperature difference (also Ma) is maintained, and changing the geometric characteristics results in discontinuous changes in the flow characteristics. Physically, this generates a $M{a}_c$ volume ratio effect, and in model learning, it is manifested as the SHAP value branch phenomenon of high Ma.

Therefore, the maximum contribution level of $V_r$ to the temperature oscillation prediction under a fixed large temperature difference (Ma = 120,000) was selected as the measurement metric, corresponding to the red dots in Fig. 11. This choice of Ma corresponds to a single observation line to see rich phenomena. At the same time, this choice does not require more prior conditions, that is, knowing $M{a}_c$, has better transferability and can better reflect the superiority of the physical interpretation of this process.

This Ma value can only be achieved in experiments with long bridges (H = 14–18 mm). The yellow points in Fig. 11 mark the maximum $\left|{\varphi }_i\right|$ corresponding to a Ma of 120,000. The relationship between Ma and $\left|{\varphi }_i\right|$ is not continuous or gradual in $V_r$ = 0.67, with two discontinuity positions indicated by the yellow points. When these points are connected by lines as shown in the figure, two distinct trajectories on the left and right branches become visible. The form of these curves is almost identical to the direct analysis of the space experiments³⁸. Furthermore, if using ${\mathrm{\varphi }}_i$ corresponding to Ma = 60,000 as the metric, the main discontinuity appears between $V_r$ values of 0.55 and 0.69. As the volume ratio decreases, the gap in ${\mathrm{\varphi }}_i$ widens, precisely corresponding to the phenomenon that the longer the mixed oscillation stage is, the smaller the volume ratio is³⁸.

The influence of the temperature gradient is examined. Figure 12 shows the relationship between the temperature gradient and the SHAP values of the volume ratio. Figures 12 and 11 exhibit the same trend that the main discontinuity appears between $V_r$ values of 0.55 and 0.69, and the gap in ${\mathrm{\varphi }}_i$ widens as $V_r$ decreases. And between $V_r$ values of 0.9 and 1.0, discontinuity represents data under different H.

Fig. 12.

The SHAP value graph of the Vr. The plot shows how the SHAP values vary with Vr, with the color gradient from red to blue indicating an increasing temperature gradient.

The analysis is the influence of $V_r$ on predicting temperature oscillations under a large Ma or temperature gradient, which has a different physical nature from the volume ratio effect on $\mathrm{\Delta }{T}_c$ or $M{a}_c$. Yet, they demonstrate a remarkable consistency. It is undeniable that the physical information is richest at the critical transition point, but the volume ratio effect persists throughout every stage of thermocapillary convection, including the point which is used above. Therefore, from the results, it can be considered that the model has learned the overall mathematical relationship of thermocapillary convection during the establishment of the temperature gradient and has established the volume ratio effect for the entire process.

From a mathematical perspective, $\mathrm{\Delta }\mathrm{T}$ : $\mathrm{grad}\mathrm{T}$: $\mathrm{M}a$ = 1 : $\frac{1}{H}$ :$cH$. The results indicate that the deep learning model is not constrained by these explicit linear equations. Within the mathematical relationships of the model high-dimensional space, all three are ultimately equivalently treated as the same kind of entity, “$\mathrm{\Delta }\mathrm{T}$”.

Another key geometric parameter is the liquid bridge height H. Figure 13 presents the ${\mathrm{\varphi }}_i$ graph of the H and Ma, with the color scale from red to blue indicating an increasing Ma. The yellow dots and lines represent the max $\left|{\mathrm{\varphi }}_{\mathrm{i}}\right|$ corresponding to the maximum Ma. When H = 14–18 mm, Ma = 120,000. The branch is already evident at H = 14, so the else part serves as an illustration suggesting a possible complete two-branch form.

Fig. 13. The SHAP value graph of the H and Ma.

The yellow dots and lines represent the maximum $\left|{\mathrm{\varphi }}_{\mathrm{i}}\right|$ corresponding to the maximum Ma.

The analysis is the influence of H on predicting temperature oscillations under a relatively large Ma. The ${\mathrm{\varphi }}_i$ values in Fig. 13 are negative, with a distinct jump occurring at H = 14 mm, serving as a node dividing the curve into two branches. Interestingly, when taking $\left|{\mathrm{\varphi }}_{\mathrm{i}}\right|$, the discontinuity between these two branches, their relative positions, and curvature are all consistent with the direct analysis of $H-\Delta T_c$ from space experiments³⁸. Another discontinuity captured at H = 10 mm and Ma = 80,000 also finds its counterpart in the direct analysis of space experiments.

Enlightening insights

The rediscovery of the Geometry Effect serves an extra significance. It validates that the Vortexformer-SHAP framework is capable of uncovering nonlinear physical regularities. Together, these aspects enhance the credibility of the enlightening insights derived in this part. For the purpose of comparison, the coordinates in this section are defined under the Input Normalization as described in the “Data preparation” section.

Figure 14 presents the bee swarm plot of ${\mathrm{\varphi }}_i$ for temperature T₁, depicting the relative importance of various input features in the model predictions. The most influential features are the temperatures at the previous time step from all five measurement points, which exhibit significantly higher ${\mathrm{\varphi }}_i$. This outcome confirms that the model operates as a spatiotemporal predictor, utilizing historical temperature data from multiple spatial locations to forecast subsequent thermocouple readings. A noteworthy finding is the unexpectedly high importance of the lower bridge temperature, which ranks second among all input features. In contrast, the SHAP values of other features, though varying in sign, remain consistently low, indicating their limited contribution to the predictions.

Fig. 14. Bee swarm plot of ${\mathrm{\varphi }}_i$ for T₁.

SHAP bee swarm plot for temperature T1. The plot displays the SHAP values for the flow temperature at point T1 across different features, illustrating the importance of each feature.

Figure 15 further visualizes the prediction of T₁ using temperatures T₂ through T₅. The lower bridge temperature leads to a distinct bifurcation in ${\mathrm{\varphi }}_i$ patterns.

Fig. 15. Using T_2–5 to predict T₁.

The Lower Bridge Temp caused the ${\mathrm{\varphi }}_i$ bifurcation. A Flow Temp point2, B Flow Temp point 3, C Flow Temp point 4, and D Flow Temp point 5.

By examining more features, it was found that the lower bridge temperature is a key bifurcation parameter in the interaction of multiple physical quantities. As shown in Fig. 16, this bifurcation phenomenon is consistently observed in the SHAP graphs of internal temperature gradient, Lyapunov exponent, average temperature, and Marangoni number, where all exhibit distinct bifurcation structures correlated with lower bridge temperature values. This systematic bifurcation across diverse flow characteristics, from thermal gradients to chaotic behavior indicators to overall flow intensity. The widespread nature of this bifurcation effect indicates that lower bridge temperature governs a fundamental continuous transition in the underlying flow physics, potentially distinguishing between different convection modes or stability states in the thermocapillary system.

Fig. 16. SHAP interaction analysis between the lower bridge temperature and normalized key system parameters.

A Internal temperature gradient, B Lyapunov exponent, C average temperature, and D Marangoni number.

Why is the lower bridge temperature so important? One possible explanation is that the model has captured the details of the lower bridge temperature adjustment from ambient to the controlled setpoint during the initial experimental phase. However, this fails to explain why the SHAP values for the upper bridge temperature, which evidently contains much more information, are significantly lower. Before a deeper analysis, the upper bridge temperature effect was first inspected. As shown in Fig. 17, its ${\mathrm{\varphi }}_i$ exhibit chaotic patterns. Through a systematic analysis of the features determining this branching behavior, four key characteristics were identified, constituting the primary factors controlling the branching structure: Lower Bridge Temperature, Height, Volume Ratio, and Entropy. Their interaction with the upper bridge temperature creates the observed multi-loop patterns. With the exception of Entropy, these are also the key parameters highlighted earlier. The upper bridge temperature indeed acts as a secondary feature in the model. When it participates in predictions alongside other variables, its own SHAP distribution and influence patterns are essentially regulated and governed by other key parameters.

Fig. 17. SHAP interaction analysis between the upper bridge temperature and four key parameters.

A Lower bridge temperature, B liquid bridge height, C entropy, and D volume ratio.

Based on the above information, a qualitative analysis from experimental and physical perspectives is as follows:

First, in the TG-2 experiments, the lower bridge temperature was typically set 2 °C below the ambient temperature, while the upper bridge temperature was established based on a target temperature difference relative to this lower bridge temperature. This setup inherently leads to a significant difference in the bulk fluid temperature depending on the lower bridge (ambient) temperature, which consequently alters fluid properties. The temperature-dependent fluid properties affected by the bulk temperature include density, viscosity, and the surface tension coefficient. All three are functions of temperature, and the overall temperature change is equivalent to the function interval change of the three physical properties. While density changes drive buoyancy on Earth, significantly altering flow dynamics, they have no fundamental effect in microgravity. However, in microgravity, the interval change of the viscosity and surface tension coefficient functions can substantially change thermocapillary convection dynamics. Furthermore, the upper bridge temperature is primarily controlled by the heat source, with its fluctuations stemming from the heat flux generated by the heating film within the temperature sampling-control cycle. In contrast, although the lower bridge temperature is also controlled by a cooling source, its fluctuations originate from the heat flux carried by the fluid flow, making it more reflective of the actual physical processes.

As shown in Fig. 18, additional delayed sequences of T₂ with different time lags were introduced, corresponding to values at different temporal positions in the T₂ sequence, for SHAP analysis. The results also reveal that the lower bridge temperature induces significant bifurcation effects across various time delays. When using the lower bridge temperature as an indicator and based on the criterion of red points, representing a higher lower bridge temperature, and blue points, representing a lower lower bridge temperature, we tracked the temporal evolution of the system. This tracking aims to provide an intuitively understandable visualization of the system dynamics over 0 to 10 time steps.

Fig. 18. Predictions of T₁ based on T₂ with increasing time lags.

A No lag (zero delay). B A lag of one time step. C A lag of five time steps. D A lag of ten time steps.

The result shows the high-temperature curve initially occupies the upper region, descends below the low-temperature curve by the 5 time steps lag, and subsequently crosses upward again by the 10 time steps lag. Based on the trend, it can be concluded that the high-temperature curve will eventually return above the low-temperature curve, forming a complete cycle. This cycle is estimated to last approximately 12 time steps, which is also the optimal input sequence length derived by the LSTM model. Specifically, the input sequence length was determined by iterating over all possible sequence lengths in the LSTM model and selecting the length corresponding to the optimal RMSE. The corresponding optimization curve is presented in Fig. 19. This provides evidence that the model has captured an intrinsic approximately 12-time-step cyclic mechanism.

This consistency might be attributed to the influence of the input sequence length on the model training. However, setting the sequence length to 12 essentially only defines the length of an observation window. This parameter controls the range of historical data the model can access at each time step, which equals the number of past steps the prediction is based on. There is no direct injection of explicit information about a periodic law into the model, which addresses part of the potential critique.

From the fundamental principles of neural networks, different input features typically activate distinct neuronal pathways and weight combinations rather than a uniform response across the entire network. Therefore, from the perspective of the Vortexformer architecture, a comprehensive explanation is that the Feature Aggregate module, through its specific weight configuration, leads to significant differences in the magnitude of SHAP values under different feature labels. The alternating pattern of “high-temperature” and “low-temperature” lines observed in the figure visualizes the numerical outcome of the weight combinations employed by the TTA module and the decoder when processing the delayed sequences.

Integrating the above two explanations: The periodic pattern manifested by the alternating high-temperature and low-temperature lines essentially reflects a periodic change in the weight combinations applied by the model to the delayed sequences versus the original sequence, cycling through phases of “similarity – divergence – renewed similarity.” These weights are a result of the model autonomously capturing data characteristics during training. Therefore, this phenomenon reveals an inherent similarity pattern between different time-series segments, which is, in essence, a manifestation of the intrinsic dynamical periodicity of the liquid bridge system visualized by the Vortexformer-SHAP framework. Therefore, the conclusion regarding the periodic law derived from the SHAP analysis imagery and the sequence length yielding the minimum RMSE are mutually corroborative, not a case of circular argumentation.

Consequently, the periodic pattern discovered in this study and the optimal sequence length (10–15 steps) derived from the LSTM model converge on the same conclusion: there exists an intrinsic periodic law in the liquid bridge experimental data (D = 20 mm, Vr = 0.54–1.15, H = 4–22 mm) of approximately 12 time steps, corresponding to a real-time duration of 5–7 s.

Discussion

Based on the Vortexformer-SHAP integrated framework developed in this study, high-precision prediction and mechanistic insight into temperature oscillations in space-based large-Prandtl-number liquid bridge thermocapillary convection were achieved. The main conclusions are as follows:

A novel, lightweight prediction architecture specifically designed for space fluid physics is proposed. The Vortexformer model, integrating spatiotemporal attention and invariant embedding, accurately models the complex dynamical processes of space-based liquid bridge thermocapillary convection. Furthermore, constrained by distance and resources, in-situ human intervention in space is complex, and remote intervention suffers from high latency. In-situ machine control would directly benefit space production. This model achieves exceptional predictive performance with only 0.02 billion ultra-light parameters, making it particularly suitable for conditions of limited resources and computing power in space environments and missions. It holds the potential for deployment on space computing platforms for real-time process control, representing the first step towards prediction and real-time control of complex thermal-fluid systems in space.

Through SHAP analysis, utilizing the critical Ma number, the volume and height effects were rediscovered. This demonstrates that the combination of uniquely designed deep models (especially Transformers) for physical mechanisms, and SHAP analysis can uncover complex nonlinear physical laws. Building on this, the interpretable model revealed two enlightening insights worthy of further investigation:

The lower bridge temperature, which represents the temperature of the cold-end supporting rod, is a key control parameter. While most research focuses on the fluid domain, few studies have specifically highlighted the influence of heat transfer through the solid cold-end rod. From a fluid physics perspective, this finding underscores the role of solid–fluid thermal interaction in large-Pr oscillatory thermocapillary convection. The interpretable model built upon space experiment data “emphasizes” the significance of cold-end rod heat transfer; however, further numerical and theoretical studies aligned with experimental conditions are needed to validate and generalize this insight.

A universal temporal periodic law of approximately 5–7 s exists within the liquid bridge geometric range of the experiment. A temporal periodic law of the flow driven solely by thermocapillary forces was derived from measurement point data. This may correspond to the spatiotemporal characteristics of a specific scaled vortex consistent across the liquid bridge geometric parameters (D = 20 mm, Vr = 0.54–1.15, H = 4–22 mm). A feasible verification approach is to establish a three-dimensional direct numerical simulation model of the liquid bridge in experimental conditions and specifically investigate the evolution of the flow field spatiotemporal configuration within each 5–7 s time window.

In summary, this study successfully applies a research paradigm combining a custom attention-based deep model with physical interpretation in the field of fluid dynamics. A powerful predictive model, Vortexformer, was developed for space-based liquid bridge thermocapillary convection, and two novel physical insights awaiting verification were proposed from a unique perspective. The developed lightweight, high-precision model is particularly well-suited to the stringent resource constraints of the space environment, representing a crucial step towards the automation and intelligence of real-time complex thermal-fluid systems control in space.

Author contributions

Haoran Gao developed the method, completed model training, analysis, wrote the manuscript text, and prepared all the figures. Jia Wang, Qi Kang, and Li Duan had completed the space experiment and provided data. All authors reviewed the manuscript.

Data availability

The data generated and analyzed during the current study are not publicly available due to privacy restrictions, but may be available from the corresponding author on reasonable request.

Code availability

The underlying code for this study is not publicly available but may be made available to qualified researchers on reasonable request from the corresponding author.

Competing interests

The authors declare no competing interests.