Going with the flow to solve for symmetry-driven PDE dynamics with physics-informed neural networks

Abstract

We address a computational framework that integrates symmetry reduction into Physics-Informed Neural Networks (PINNs) for analysing symmetry-driven dynamics in nonlinear partial differential equations (PDEs). Using an auxiliary network to learn time-dependent transformations, we render the symmetry-invariant solutions stationary or slowly varying in rescaled coordinates while simultaneously inferring the symmetry parameters (e.g., wave speed, scaling rates). This yields a modified evolution equation coupled with algebraic constraints on symmetry parameters, producing index-2 differential-algebraic equation (DAE) systems. Since conventional standard PDE/ODE solvers struggle or even fail with such high-index DAEs, we employ PINNs as an alternative approach that naturally unifies PDE residuals and algebraic constraints in a single loss function. This allows simultaneous inference of the invariant solutions and the transformation properties without large domains, mesh adaptivity, or front tracking. Beyond forward simulation, our framework further enables robust parameter inference from sparse, spatially offset data where vanilla PINNs fail. Our numerical demonstrations include, among others, the 2D porous medium, the generalised Korteweg-de Vries and Burgers PDE, showcasing our proposed approach as a powerful tool for the solution of both the forward and inverse problems for index-2 DAEs arising in nonlinear wave and scaling dynamics.

Many nonlinear partial differential equations exhibit dynamics governed by continuous symmetries. Here, authors integrate symmetry reduction into physics-informed neural networks, enabling accurate forward simulation and parameter inference without front tracking or mesh adaptation.

Introduction

Many nonlinear partial differential equations (PDEs) modeling complex spatio-temporal dynamics across physics, biology, and engineering exhibit solutions characterized by symmetry-driven behavior, such as traveling waves, self-similar collapse, finite-time blow-up, rotating patterns, or radial expansions^1–13. These behaviors often reflect underlying continuous symmetries in the governing equations, such as translational, scaling, or rotational invariance. Exploiting these symmetries can simplify the analysis and reduce the complexity of the observed dynamics, but it typically requires reformulating the PDE by introducing additional terms—such as convective or scaling components—that account for the effects of translation, dilation, or rotation. These modifications effectively anchor the solution in a co-moving or rescaled frame, removing symmetry-induced drift or growth, but result in a transformed system that includes additional unknowns and algebraic constraints, often increasing the mathematical and numerical complexity of the problem⁶. For example, traveling wave problems can be reformulated in a moving frame with unknown speed, whereas self-similar problems may involve dynamically rescaled coordinates, often governed by parameters, such as the rate of change of the width or of the amplitude, that must be inferred as part of the solution.

To uniquely determine the transformed solution, one often supplements the reformulation with template-based conditions or additional ad hoc constraints. These may include pointwise anchoring (e.g., fixing a value or derivative at a reference location, such as an inflection point), orthogonality conditions, or minimization of integral functionals measuring the distance between the evolving solution and a prescribed template function. Such constraints facilitate isolating the intrinsic dynamics from symmetry-related modes and guide the solution toward a unique identification in the transformed frame. When applied systematically, these approaches typically yield systems of differential-algebraic equations (DAEs), where the algebraic part enforces normalization, symmetry breaking, or projection conditions tailored to the specific traveling or scaling behavior of interest⁶.

High-index (≥2) DAEs pose severe challenges for traditional numerical solvers^14–17. Such systems are frequently stiff, sensitive to initial condition consistency, and require specialized integration techniques. These difficulties are compounded in the presence of multiple time scales, conservation laws, or quasi-steady-state approximations, where standard time-stepping methods may fail to capture the correct dynamics or preserve invariants. Additionally, when dealing with singularities or long-range traveling structures, classical PDE solvers require either large computational domains or adaptive mesh refinement—both of which increase computational cost and can lead to numerical instabilities.

Recent advances in scientific machine learning have opened new pathways for solving forward^18–22 and inverse problems^23–31 of nonlinear PDEs and index-1 DAEs³². Early works in the 1990s already explored the use of neural networks for approximating solutions^33–36 and identifying system dynamics^37–42, setting the foundations for modern physics-informed approaches. Building on these ideas, physics-informed neural networks (PINNs)^43–45 have emerged as a promising framework, where the governing equations, along with initial, boundary, and possibly algebraic constraints, are embedded directly into the training of neural networks via a loss function that can incorporate sparse measurement data. This approach allows for the solution of both classical initial/boundary value problems and more complex scenarios involving data-driven calibration and/or discovery of unknown parameters and hidden dynamics, with applications across fields, such as fluid mechanics and materials science^46–51. The framework has further been extended to more general nonlinear functional equations arising in dynamical systems, including feedback linearization⁵², observer design⁵³, identification of slow-invariant manifolds in multiscale systems^54,55 and learning normal form coordinate transformations⁵⁶. Such efforts have extended past continuum problems to nonlinear dynamical lattice ones^57,58.

PINNs can be readily applied to DAEs by incorporating both differential operators and algebraic constraints directly into the loss function. This approach avoids the need for index reduction procedures, specialized DAE solvers, or dedicated time-stepping schemes^59,60. Constraints arising from template conditions, normalization requirements, or even integral identities can be enforced seamlessly as additional (soft) penalty terms in the training objective. This flexibility can be particularly useful in capturing self-similar and solitonic behaviors, where traditional numerical solvers often face difficulties due to mesh adaptivity or domain deformation requirements. Recent applications include PINN-based modeling of spontaneous imbibition and self-similar flows⁶¹, the identification of asymptotically self-similar solutions of axisymmetric Euler flows⁶², the study of optical solitons with conservation constraints⁶³, and nonlinear wave equations with adaptive training strategies⁶⁴.

In this work, we propose a computational framework that integrates symmetry reduction into the PINN formulation for the solution of both the forward and inverse problems for index-2 DAEs. Such approach is specifically tailored to solving PDEs that exhibit traveling waves, self-similarity, and collapse/blow-up behavior through transformation into an index-2 DAEs system. The methodology builds on the template-based representations^7,8,65,66 to reduce the system while retaining the essential dynamics. Although the individual components (PINNs and symmetry-reduction frameworks) exist, their integration into a unified framework for directly solving index-2 DAE systems without index reduction or specialized solvers constitutes the main contribution of the present work. The PINN architecture jointly learns the rescaled solution and the evolution of group parameters (front velocities, scaling rates). Concretely, the architecture trains two interconnected networks simultaneously: (i) a network admitting space and time as inputs to represent the rescaled solution, and (ii) an auxiliary network that takes time alone to learn time-dependent transformations, that renders the symmetry-invariant solutions stationary or slowly varying in rescaled coordinates. We demonstrate that this approach avoids the need for large computational domains or adaptive meshing by working directly in the transformed coordinates, solving the full coupled high-index DAE system using the PINN loss formulation.

Beyond the solution of the forward problem, the proposed framework naturally extends to the solution of inverse problems. By embedding sparse observational data into the loss function, our proposed PINN framework can simultaneously learn the reduced solution, the symmetry parameters, and unknown physical coefficients. In this way, the dynamically reconstructed co-moving frame aligns the front with the sensor, enabling accurate calibration even with delayed, spatially offset measurements. Moreover, the framework is further extended to a steady-state symmetry-reduced formulation that directly computes self-similar blow-up solutions by treating the blow-up rate as a trainable parameter. This enables, for example, the study of singularity-forming PDE dynamics through their associated self-similar stationary structures.

We validate our framework across several representative examples: (i) the Nagumo PDE, which exhibits traveling fronts; (ii) the linear diffusion PDE in two dimensions with self-similar decay; (iii) the 2D porous medium equation exhibiting self-similarity and nonlinear spreading; (iv) the Burgers PDE, combining both traveling and scaling features; and (v) the generalized Korteweg-de Vries (gKdV) PDE with self-similar blow-up solutions. Additionally, the 1D diffusion and the axisymmetric 2D porous medium PDE benchmarks are presented in the Supplementary Material (Sections 1 and 2). These cases illustrate the generality of the approach and highlight the potential of PINNs to serve as a robust tool for analysing symmetry-driven PDE dynamics that naturally lead to high-index DAE formulations. Moreover, by transforming to a frame where the solution is nearly stationary, our framework reduces stiffness and simplifies the learning task, thereby improving PINN performance on problems that would otherwise require adaptive meshes or front tracking.

In the text, we will use ∂_x to denote the partial derivative $\frac{\partial }{\partial x}$, for notational convenience.

Results

In this section, we showcase the application of the PINN-assisted MN framework across five distinct case studies: one involving a traveling wave solution (Nagumo), three exhibiting self-similar behavior (the 2D diffusion, the 2D porous medium, and the generalized Korteweg-de Vries PDE), and one combining both scaling and translational invariance (Burgers PDE). For each case, we formulate an index-2 DAE system and demonstrate how the PINN-assisted methodology can be used to learn both the invariant (translational or self-similar) profile and the corresponding wave speed and/or self-similar exponents within a dynamically rescaled coordinate system. All simulations were performed in MATLAB on a laptop with an 11th Gen Intel(R) Core(TM) i7-111800H @2.30CPU GHz processor, 32 GB RAM, and Windows 11 Home operating system. The neural networks were implemented using dlnetwork and dlarray objects to enable automatic differentiation. The architecture of the neural networks follows a simple feedforward structure with hyperbolic tangent activations. Each model is defined using the functions featureInputLayer, fullyConnectedLayer, and tanhLayer. Training was performed using a quasi-Newton method (L-BFGS) via the lbfgsupdate routine, suitable for low-batch, PINN settings.

Remark

As a practical strategy to improve convergence, we introduced a warm-up phase prior to full training. Specifically, the network is first trained to approximate the initial condition alone, extended trivially over all times (i.e., enforcing that the initial condition remains constant across time). This preconditioning step facilitates the network in establishing a suitable parameter initialization before minimizing the full physics-informed loss, leading to more stable convergence during the subsequent optimization. No further training complexities (such as adaptive loss weighting, greedy training or domain decomposition) were needed, as the symmetry reduction already regularizes and reduces the stiffness of the PDE problem.

Case study 1: traveling wave in the Nagumo PDE

We first illustrate the proposed methodology using a translationally invariant problem: the Nagumo equation, given by

$${\partial }_{t}u={\partial }_{xx}^{2}u+u\left(1-u\right)\left(u-a\right).$$ 1

The Nagumo equation is a classical example of a parabolic PDE that supports traveling wave solutions.

An explicit traveling wave solution takes the form u(x, t) = w(x − c), where:

$$w\left(x\right)={\left[1+\exp \left(-\frac{x}{\sqrt{2}}\right)\right]}^{-1},\frac{\mathrm{d}c}{\mathrm{d}t}=-\sqrt{2}\left(\frac{1}{2}-a\right).$$ 2

To capture the traveling wave behavior, we dynamically construct a co-moving frame, with coordinate y ≡ x − c(t). In this framework, we consider a solution of the form u(x, t) = w(x − c(t), t), which substituted into the translational invariant PDE in Eq. 1, we obtain the modified PDE:

$${\partial }_{t}w={\partial }_{yy}^{2}w+w\left(1-w\right)\left(w-a\right)-\frac{\mathrm{d}c}{\mathrm{d}t}{\partial }_{y}w.$$ 3

The instantaneous wave speed dc/dt is an additional degree of freedom, closed by an algebraic-template condition that anchors the solution in the co-moving frame to a predefined template function T(y):

$${\int }_{y_{\min }}^{y_{\max }}\left(w\left(y,t\right)-T\right){\partial }_{y}T\mathrm{d}y=0.$$ 4

Here, we solve Eq. 3 in the domain y ∈ [−30, 30] with Neumann boundary conditions, parameter a = 0.01, and initial condition

$$w\left(y,0\right)=\left\{\begin{array}{cc}0,\hfill & y<0\hfill \\ y/10,\hfill & 0\le y\le 10\hfill \\ 1,\hfill & y>10\hfill \end{array}\right),$$ 5

which also serves as the template function T(y) = w(y, 0). We employ two neural networks: ${\mathcal{N}}_w\left(y,t\right)$ (two hidden layers, 20 neurons each) approximates w(y, t), and ${\mathcal{N}}_p\left(t\right)$ (one hidden layer, 5 neurons) approximates the instantaneous wave speed dc/dt.

Both networks use the hyperbolic tangent as the activation function. Training is performed using the L-BFGS optimizer, which ensures efficient convergence of the total loss ${\mathcal{E}}_{total}$ (see Fig. 1a). For ${\mathcal{N}}_w\left(y,t\right)$, we use 119,800 collocation in the y − t domain (599 equidistributed points in y ∈ (−30, 30) and 200 equally distributed points in t ∈ [(0, 20)]. For ${\mathcal{N}}_p\left(t\right)$, we use 201 equally spaced points in t ∈ [0, 20]. The spatio-temporal solution is depicted in Fig. 1b. This closely matches the theoretical value $\sqrt{2}\left(\frac{1}{2}-0.01\right)=0.692965$. As depicted in Fig. 1d, the template condition enforces the solution w(y, t) in the co-moving frame to become stationary in shape, converging to a traveling wave form. In addition, the comparison between w at long time (t = 20) and the theoretical profile (Eq. 2) shows excellent agreement.

Fig. 1. PINN prediction of the MN dynamics for the co-moving Nagumo equation.

a Convergence of the total loss ${\mathcal{E}}_{\mathrm{total}}$ during training. b Evolution of the rescaled solution w(y, t) in co-moving coordinates. c Predicted instantaneous velocity $\frac{\mathrm{d}c}{\mathrm{d}t}$ for a = 0.01, approaching the theoretical value, $\sqrt{2}\left(\frac{1}{2}-a\right)$ (black dashed-dotted line). d Snapshots of the rescaled (co-moving) solution w at various time instances (t = 0.5: red line with circles, t = 2: blue line with circles, t = 20: crossed black line). At long times (t > 10), the PINN-predicted solution w converges to a stationary profile that shows excellent agreement with the theoretical expression (green dashed line) in Eq. 2, up to a spatial shift of approximately 5 units in x.

For comparison, we also solve Eq. 3 using central finite differences (FD) with backward Euler time stepping. For this simple problem, the conventional solver obtains similar results to the PINN at a substantially lower wall-clock cost (~15 s vs. ~80 min for PINN training), underscoring that PINNs are not necessarily advantageous when efficient standard numerical methods are available⁶⁷. This comparison is presented in Section 3.1 of the Supplementary Material.

Parameter inference in the co-moving frame

An important extension of MN dynamics, when combined with PINNs, is the ability to accurately infer unknown parameters of the governing PDE, even when observational data are collected far from the traveling front. To illustrate this capability, we consider a modified version of Eq. 1:

$${\partial }_{t}u=D{\partial }_{xx}^{2}u+u\left(1-u\right)\left(u-a\right),$$ 6

where D is the unknown parameter to be inferred and a = 0.01.

To generate synthetic data, Eq. 6 is solved using a finite-element method (FEM), and the solution u is recorded at a fixed “sensor” location x^* = − 10 and at times t^* = 2, 4, 6, 8, 10. In this example, the reference solution is computed with D = 2. Notably, the sensor location x^* lies far from the propagating front during the early stages (t < 10), and only at the latest time t^* = 10 does the sensor detect a non-negligible signal, u(x^* = − 10, t^* = 10) ≈ 0.03.

We compare the standard (“vanilla”) PINN against our co-moving PINN formulation for parameter inference. The vanilla PINN $\mathcal{N}\left(x,t\right)$ (two hidden layers, 20 neurons each, tanh activation) is trained with L-BFGS for 10,000 epochs using 119,800 collocation points, with a loss that includes PDE, initial, boundary, and sparse data terms. It infers $\widehat{D}=1.76$, which deviates substantially from the true value D = 2 (~10% error) and fails to accurately reconstruct the solution at late times (Fig. 2a, b). In contrast, our co-moving PINN formulation determines dc/dt via the template condition in Eq. 4. With the same architectures and training settings, this co-moving formulation yields $\widehat{D}=1.978$ (~1% error), accurately predicts the asymptotic wave speed (dc/dt ≈ 0.979 vs. theoretical 0.98), and maintains accuracy at late times where the vanilla PINN deteriorates substantially (Fig. 2a, b).

Fig. 2. Comparison of the vanilla PINN and the PINN + MN solutions for the co-moving Nagumo equation.

Solution profiles at a t = 10 and b t = 20, compared against the reference solution obtained using the finite element method (dashed red line). Both PINN models are trained to infer D using data collected at x^* = − 10 and times t^* = 2, 4, 6, 8, 10. The vanilla PINN (solid line with rectangles) predicts $\widehat{D}=1.76$, whereas the PINN combined with MN dynamics (solid line with triangles) yields $\widehat{D}=1.98$.

Case study 2: self-similarity in the 2D diffusion PDE

As a second illustrative example, we consider the two-dimensional diffusion equation:

$${\partial }_{t}u={\partial }_{xx}^{2}u+{\partial }_{yy}^{2}u,$$ 7

the right-hand side operator ${\mathcal{D}}_{\mathbf{x}}u\equiv {\partial }_{xx}^{2}u+{\partial }_{yy}^{2}u$ satisfies the scaling relation:

$$\begin{array}{c}{\mathcal{D}}_{\mathbf{x}}\left(Bw\left(\frac{x}{A},\frac{y}{A}\right)\right)=B{A}^{-2}{\mathcal{D}}_{\mathit{\xi }}w,\hfill \\ \mathrm{with}\mathit{\xi }\equiv \left(\xi ,\eta \right)=\left(\frac{x}{A},\frac{y}{A}\right).\hfill \end{array}$$ 8

In our dynamic renormalization framework, discussed earlier, e.g., in ref. ⁶⁸, we introduce the ansatz:

$$u\left(x,t\right)=B\left(\tau \right)w\left(\frac{x}{A\left(\tau \right)},\frac{y}{A\left(\tau \right)},\tau \left(t\right)\right),$$ 9

and substituting in Eq. 7 using Eq. 8 boils down to:

$${\partial }_{\tau }w+\frac{{\partial }_{\tau }B}{B}w-\frac{{\partial }_{\tau }A}{A}\left(\xi {\partial }_{\xi }w+\eta {\partial }_{\eta }w\right)={\partial }_{\xi \xi }^{2}w+{\partial }_{\eta \eta }^{2}w,\mathrm{with}\xi =\frac{x}{A\left(\tau \right)},\eta =\frac{y}{A\left(\tau \right)},\mathrm{and}{\partial }_t\tau =A^{-2}.$$ 10

Notice that here, given the isotropy of the spatial operator, we have assumed that the two spatial directions scale in the same way. We determine the scaling rates, $G\left(\tau \right)\equiv \frac{{\partial }_{\tau }B}{B}$ and $C\left(\tau \right)\equiv \frac{{\partial }_{\tau }A}{A}$ by imposing two algebraic template conditions. These conditions arise from minimizing the functional:

$$E\equiv {\int }_{{\eta }_{\min }}^{{\eta }_{\max }}{\int }_{{\xi }_{\min }}^{{\xi }_{\max }}{\left(w-BT\left(\frac{x}{A},\frac{y}{A}\right)\right)}^2\mathrm{d}x\mathrm{d}y,$$ 11

when A = B = 1. This leads to the following constraints:

$${\int }_{{\eta }_{\min }}^{{\eta }_{\max }}{\int }_{{\xi }_{\min }}^{{\xi }_{\max }}\left(w-T\right)T\mathrm{d}\xi \mathrm{d}\eta =0\mathrm{and}$$ 12

$$\begin{array}{c}{\int }_{{\eta }_{\min }}^{{\eta }_{\max }}{\int }_{{\xi }_{\min }}^{{\xi }_{\max }}\left(w-T\right)\left(\xi {\partial }_{\xi }T+\eta {\partial }_{\eta }T\right)\mathrm{d}\xi \mathrm{d}\eta =0,\hfill \end{array}$$ 13

with T(ξ, η) a template function. For our illustrative example, we solve Eq. 10 with initial condition (which also serves as template condition):

$$w\left(\xi ,\eta ,0\right)=T\left(\xi ,\eta \right):=\exp \left(-\left(\mid \xi \mid +\mid \eta \mid \right)\right)$$ 14

in the domain (ξ, η) ∈ (0, 4) × (0, 4) with symmetry boundary conditions at ξ = 0, η = 0, and Dirichlet boundary conditions, w(ξ, η, τ > 0) = 0 at boundaries ξ = 4 and η = 4. The neural network ${\mathcal{N}}_w\left(\mathit{\xi },\tau \right)$ consists of three hidden layers with 40 neurons each, and uses hyperbolic activation functions. The auxiliary network ${\mathcal{N}}_p\left(\tau \right)$ for the prediction of scaling rates, G(τ) and C(τ) has a single hidden layer with five neurons. Training is performed using the L-BFGS optimizer. The neural network ${\mathcal{N}}_w\left(\mathit{\xi },\tau \right)$ is trained using 70,321 collocation points in the rescaled spatial-temporal domain.

Figure 3a–c depict snapshots of the rescaled solution w at rescaled τ = 0, 0.5, and 3. The neural network predicts that the rescaled solution w practically converges to a stationary self-similar profile after τ ≈ 3. In addition, Fig. 3d shows the ${\mathcal{N}}_p\left(\tau \right)$–based predicted evolution of the ratio of scaling rates $G/C\equiv \frac{{\partial }_{\tau }B/B}{{\partial }_{\tau }A/A}$, which converges to a stationary value of approximately −2. In particular, the scaling rates stabilize at steady-states (ss) G_ss = − 2.0236 and C_ss = 1.0078, and their ratio allows estimation of the self-similar exponents of the 2D diffusion equation.

Fig. 3. PINN prediction of the MN dynamics for the 2D diffusion equation.

a–c Snapshots of the rescaled solution w(ξ, η, τ) at rescaled times τ = 0, 0.5, and 3. The rescaled solution converges to a stationary, self-similar profile for τ > 3. The neural network ${\mathcal{N}}_w\left(\mathit{\xi },\tau \right)$ is trained using 70,321 collocation points in the rescaled spatial-temporal (ξ, η) − τ domain. d Evolution of scaling rates G/C ratio over rescaled time, τ. As τ → ∞, the ratio G/C converges to a steady-state value, ${lim}_{\tau \to \infty }G/C=-2$, which is used to derive the self-similar exponents of the diffusion equation.

From ∂_τB/B = G_ss and ∂_τA/A = C_ss, $B=B_0{e}^{G_{ss}\tau }$ and $A=A_0{e}^{C_{ss}\tau }$. Through the last of Eq. 10 leads to $e^{C_{ss}\tau }=\frac{\sqrt{2C_{ss}}}{A_0}{\left(t+t_0\right)}^{1/2}$, where t₀ is a non-negative integration constant. Using this derived transformation between τ and the original time t in the equations for B and A, we infer that $A\propto {\left(t+t_0\right)}^{1/2}$, and $B\propto {\left(t+t_0\right)}^{G_{ss}/\left(2C_{ss}\right)}$ which leads to an amplitude exponent of ≈ − 1.

Case study 3: self-similar solutions of the 2D porous medium equation

We now examine the application of MN dynamics, combined with PINNs, to identify self-similar solutions and corresponding self-similar exponents of the 2D porous medium equation:

$${\partial }_{t}u={\partial }_{rr}^2{u}^2+\frac{1}{r}{\partial }_r{u}^2+\frac{1}{r^2}{\partial }_{\theta \theta }^2{u}^2.$$ 15

expressed in polar coordinates, (r, θ). The nonlinear operator ${\mathcal{D}}_{r\theta }u\equiv {\partial }_{rr}^2{u}^2+\frac{1}{r}{\partial }_r{u}^2+\frac{1}{r^2}{\partial }_{\theta \theta }^2{u}^2$ satisfies the scaling relation:

$${\mathcal{D}}_{r\theta }\left(Bw\left(\frac{r}{A},\theta \right)\right)=B^2{A}^{-2}{\mathcal{D}}_{\rho \theta }w,\mathrm{with}\rho =\frac{r}{A}.$$ 16

Using the scaling ansatz:

$$u\left(r,\theta ,t\right)=B\left(\tau \right)w\left(\frac{r}{A\left(\tau \right)},\theta ,\tau \left(t\right)\right)$$ 17

and substituting into Eq. 15 yields the MN-transformed system:

$${\partial }_{\tau }w-\frac{{\partial }_{\tau }B}{B}w+\frac{{\partial }_{\tau }A}{A}\rho {\partial }_{\rho }w={\partial }_{\rho \rho }^2{w}^2+\frac{1}{\rho }{\partial }_{\rho }{w}^2+\frac{1}{{\rho }^2}{\partial }_{\theta \theta }^2{w}^2,\mathrm{with}{\partial }_t\tau =B{A}^{-2}.$$ 18

The unknown scaling rates G(τ) ≡ ∂_τB/B and C(τ) ≡ ∂_τA/A are determined through two template (algebraic) constraints following the dynamic renormalization framework of ref. ⁶. Because the renormalised formulation is invariant under certain rescalings, these constraints act as gauge conditions that select a unique representative of the self-similar solution. The first constraint is an orthogonality condition:

$${\int }_{{\theta }_{\min }}^{{\theta }_{\max }}{\int }_{{\rho }_{\min }}^{{\rho }_{\max }}w\left(\rho ,\theta \right)T\left(\rho \right)\mathrm{d}\rho \mathrm{d}\theta =0,\mathrm{where\; :}T\left(\rho \right)=1-\frac{2}{1+\exp \left(-5\left(\rho -1\right)\right)}$$ 19

is a smooth step-like template. This template is not unique; similar smooth functions could be used to fix the spatial scaling freedom with the same asymptotic self-similar state. The second constraint is a pinning condition fixing the amplitude at ρ = 0:

$$w\left(0,\theta ,\tau \right)=1.$$ 20

The computational domain is (ρ, θ) ∈ (0, 4) × (0, π/2), with boundary conditions: ∂_ρu = 0 at ρ = 0, w(4, τ) = 0, and symmetry conditions, ∂_θu = 0 at ${\theta }_{\min }=0$ and ${\theta }_{\max }=\pi /2$.

The initial condition is:

$$w\left(\rho ,\theta ,0\right)=\frac{1}{1+\exp \left[-5\left(2-\rho +0.3\cos \left(2\theta \right)\right)\right]}.$$ 21

The resulting system of DAEs is solved using the proposed MN-PINN framework. The network ${\mathcal{N}}_w\left(\rho ,\theta ,\tau \right)$ consists of three hidden layers with 40 neurons each and employs the hyperbolic tangent activation function, whereas a second network ${\mathcal{N}}_p\left(\tau \right)$ predicts G(τ) = ∂_τB/B and C(τ) = ∂_τA/A. Training uses 85,571 collocation points and minimizes a composite loss function including PDE, boundary, initial and algebraic constraint terms. Optimization is performed using L-BFGS for 30,000 epochs (this corresponds to a total computational time of ~150 min).

As depicted in Fig. 4a–c, the initially non-radially symmetric profile rapidly evolves toward a radially symmetric self-similar solution. This demonstrates that the MN-PINN framework not only captures the correct asymptotic state but also resolves the symmetry restoration inherent to the dynamics. The scaling exponents are extracted for the steady-state values of the G and C rates. Evolving the system to τ = 6 yields an approximation of steady-states (ss) with G_ss ≈ − 1.035 and C_ss ≈ 0.5195. These imply $A\propto {\left(t+t_0\right)}^{C_{ss}/\left(2C_{ss}-G_{ss}\right)}\approx {\left(t+t_0\right)}^{0.251}$, and $B\propto {\left(t+t_0\right)}^{G_{ss}/\left(2C_{ss}-G_{ss}\right)}\approx {\left(t+t_0\right)}^{-0.498}$, with t₀ an integration constant. These values are in excellent agreement with the theoretical exponents α = 0.25 and β = − 0.5, confirming that the proposed PINN-assisted MN framework accurately captures both the self-similar solution and its scaling laws for the 2D porous medium equation.

Fig. 4. PINN prediction of the MN dynamics for the 2D porous medium equation.

a–c Snapshots of the rescaled solution w(ρ, θ, τ) at (rescaled) τ = 0, 0.5, and 3. The solution converges rapidly to a stationary, self-similar profile for τ ≳ 2. Insets illustrate top views, highlighting the transition from a non-radially symmetric initial condition to a radially symmetric self-similar solution. (d) Evolution of the scaling rates C (solid line with circles) and G (solid line with triangles) as a function of τ. As τ → ∞, the ratio converges to ${lim}_{\tau \to \infty }G/C\approx -1.99$, which determines the self-similar exponents.

We note that, for this particular case, conventional numerical methods (see Supplementary Material Section 3.3) failed to complete the simulation, terminating prematurely because of growing numerical instabilities. Even at substantially higher spatio-temporal resolution, a finite-element solver with backward-Euler time stepping could not accurately track the developing front of the rescaled solution w (see Supplementary Fig. 5). On the other hand, our proposed MN-PINN model completed training and produced accurate estimates of the self-similar profile and the relevant scaling exponents, without requiring specialized numerical schemes or adaptive-meshing.

Case study 4: a problem with scale and translational invariance—the Burgers equation

Consider the one-dimensional Burgers equation:

$${\partial }_{t}u=\nu {\partial }_{xx}^{2}u-u{\partial }_{x}u\equiv {\mathcal{D}}_x\left(u\right).$$ 22

Here, the value of viscosity is set ν = 0.025. We apply the scaling ansatz:

$$u\left(x,t\right)=Bw\left(\frac{x-c}{A}\right),$$ 23

which yields the following relation for the operator ${\mathcal{D}}_x$:

$$\begin{array}{ccc}{\mathcal{D}}_x\left(Bw\left(\frac{x-c}{A}\right)\right)\hfill & =\hfill & B{A}^{-2}\nu {\partial }_{yy}^{2}w-B^2{A}^{-1}w{\partial }_{y}w,\hfill \\ \hfill & \hfill & \mathrm{with}y=\frac{x-c}{A}.\hfill \end{array}$$ 24

This means that the operator ${\mathcal{D}}_x$ is invariant under the transformation when the condition AB = 1 holds. Under this scaling, we obtain:

$${\mathcal{D}}_x\left(\frac{1}{A}w\left(\frac{x-c}{A}\right)\right)=A^{-3}{\mathcal{D}}_y\left(w\right).$$ 25

Now, substituting the dynamic scaling form:

$$u\left(x,t\right)=\frac{1}{A\left(\tau \right)}w\left(\frac{x-c\left(\tau \right)}{A\left(\tau \right)},\tau \left(t\right)\right)$$ 26

into Eq. 22 leads to the dynamically rescaled equation:

$${\partial }_{\tau }w=\nu {\partial }_{yy}^{2}w-w{\partial }_{y}w+\frac{{\partial }_{\tau }A}{A}\left(w+y{\partial }_{y}w\right)+\frac{{\partial }_{\tau }c}{A}{\partial }_{y}w,\mathrm{with}y=\frac{x-c}{A}.$$ 27

The two τ-dependent unknown quantities, ∂_τA/A and ∂_τc/A are determined by imposing two template constraints. These arise by minimizing:

$$E\equiv {\int }_{y_{\min }}^{y_{\max }}{\left(w-\frac{1}{A}T\left(\frac{y-c}{A}\right)\right)}^2\mathrm{d}y,$$ 28

evaluated at A = 1 and c = 0. T(y) is a chosen template function. This leads to the following constraints:

$${\int }_{y_{\min }}^{y_{\max }}\left(w-T\right)\left(T+y{\partial }_{y}T\right)\mathrm{d}y=0,\mathrm{and}{\int }_{y_{\min }}^{y_{\max }}\left(w-T\right){\partial }_{y}T\mathrm{d}y=0.$$ 29

Here, we select $T\left(y\right)=\exp \left(-y^2\right)$. Equation (27) is solved in the domain y ∈ [−6, 6] with zero-flux boundary conditions at both ends. The initial condition is $w\left(y,0\right)=\exp \left(-y^2\right)$.

For this implementation, the neural network ${\mathcal{N}}_w\left(y,\tau \right)$ consists of three hidden layers with 20 neurons each, using the hyperbolic tangent activation function. The auxiliary network ${\mathcal{N}}_p\left(\tau \right)$ which predicts ∂_τA/A and ∂_τc/A contains one hidden layer with 4 neurons. The neural network ${\mathcal{N}}_w\left(y,\tau \right)$ is trained using 179,700 collocation points in the (y, τ) domain. Training is performed using the L-BFGS optimizer, and the convergence of the total loss ${\mathcal{E}}_{total}$ is shown in Fig. 5d. Although the initial data steepens into a sharp front, the front neither propagates nor increases in amplitude—unlike the solution of the original Burgers equation in the original (x, t)—domain. Indeed, this demonstrates that we have converged to a stationary (indeed, stable, since it was reached through evolution dynamics) self-similar traveling solution. To evaluate the accuracy of our PINN-assisted framework, we compare the rescaled solution w(y, τ) at τ = 6—where the solution has effectively converged—with the analytical self-similar solution of the Burgers equation²:

$$u\left(x,t_0\right)=\sqrt{\frac{\nu }{\pi t^{*}}}\frac{\left[\exp \left(\frac{A^{*}}{2\nu }\right)-1\right]\exp \left[\frac{-{\left(x-c^{*}\right)}^2}{4\nu t^{*}}\right]}{1+\left[\frac{1}{2}\exp \left(\frac{A^{*}}{2\nu }\right)-\frac{1}{2}\right]\cdot \mathrm{erfc}\left(\frac{\left(x-c^{*}\right)}{\sqrt{4\nu t^{*}}}\right)}$$ 30

The evolution of ∂_τA/A and ∂_τc/A over rescaled time τ is shown in Fig. 5b. As depicted in Fig. 5c, there is excellent agreement between the theoretical self-similar solution and the PINN-predicted profile w, with optimal parameters: amplitude, A^* ≈ 1.7713, spatial shift: c^* ≈ − 1.8715 and time shift: t^* ≈ 2.1666.

Fig. 5. PINN prediction of the MN dynamics for the Burgers equation.

a Spatio-temporal evolution of the dynamically rescaled solution w(y, τ). For long τ values (τ > 3), the rescaled solution w converges to a stationary, self-similar profile. b Evolution of ∂_τA/A (line with circles) and ∂_τc/A (line with squares) over rescaled time τ. c Snapshots of rescaled PINN predicted solution w at different τ values (τ = 0.1: black line with circles, τ = 1: green line with triangles, τ = 6: blue line with squares). The converged self-similar solution w (after practically τ ≈ 3) and the theoretical self-similar solution (red dashed line)² show excellent agreement. d Convergence of the total loss ${\mathcal{E}}_{total}$ during training for the rescaled/co-traveling Burgers equation. The neural network ${\mathcal{N}}_w\left(y,\tau \right)$ is trained using 179,700 collocation points in the y − τ domain.

Case study 5: a steady-state approach for computing self-similar blow-up solutions in the Korteweg-de Vries problem

In this subsection, we extend the MN dynamics framework to compute self-similar blow-up solutions by directly formulating and solving the corresponding steady-state problem. We consider the generalized Korteweg-de Vries (gKdV) equation:

$${\partial }_{t}u=-{\partial }_{xxx}u-{\partial }_x{u}^p\equiv {\mathcal{D}}_x\left(u\right),$$ 31

where the nonlinearity exponent p ≥ 2 controls the solution behavior. In particular, for p > 5, the equation enters the supercritical regime, where solutions exhibit finite-time blow-up.

The gKdV equation admits traveling wave solutions of the form:

$$u\left(x,t\right)={\left(\frac{c\left(p+1\right)}{2}\right)}^{\frac{1}{p-1}}{\mathrm{sech}}^{\frac{2}{p-1}}\left(\frac{\sqrt{c}\left(p-1\right)}{2}\xi \right)$$ 32

where ξ: = x − ct, and c is the wave speed. These solutions become linearly unstable for p ≥ 5, signaling the onset of blow-up dynamics⁶⁹. To capture self-similar blow-up behavior, we introduce the scaling ansatz:

$$\begin{array}{ccc}u\left(x,t\right)\hfill & :=\hfill & B\left(\tau \right)w\left(\frac{x}{A}\left(\tau \right)+K\left(\tau \right),\tau \right),\hfill \\ \xi \hfill & :=\hfill & \frac{x}{A\left(\tau \right)}+K\left(\tau \right),\tau :=\tau \left(t\right),\hfill \end{array}$$ 33

where A(τ), B(τ), and K(τ) represent the evolving width, amplitude, and spatial shift, respectively. The nonlinear operator satisfies the scaling relation:

$${\mathcal{D}}_x\left(Bw\left(\frac{x}{A}+K\right)\right)=-B{A}^{-3}{\partial }_{\xi \xi \xi }w-B^p{A}^{-1}{\partial }_{\xi }{w}^p.$$ 34

Scale invariance is achieved under the constraint B = A^−2/(p−1), yielding:

$${\mathcal{D}}_x\left(A^{-2/\left(p-1\right)}w\left(\frac{x}{A}+K\right)\right)=A^{\frac{1-3p}{p-1}}{\mathcal{D}}_{\xi }\left(w\right).$$ 35

Substituting the ansatz into Eq. 31 leads to the MN-transformed system:

$$\begin{array}{c}{\partial }_{\tau }w=-{\partial }_{\xi \xi \xi }w-{\partial }_{\xi }{w}^p+\frac{{\partial }_{\tau }A}{A}\left(\frac{2}{p-1}w+\xi {\partial }_{\xi }w\right)\hfill \\ -\left(\frac{{\partial }_{\tau }A}{A}K+{\partial }_{\tau }K\right){\partial }_{\xi }w,\hfill \end{array}$$ 36

and imposing the condition $\frac{{\partial }_{\tau }A}{A}K+{\partial }_{\tau }K=-1$ reduces the dynamics to a co-moving frame with unit wave speed, yielding:

$${\partial }_{\tau }w=-{\partial }_{\xi \xi \xi }w-{\partial }_{\xi }{w}^p+\frac{{\partial }_{\tau }A}{A}\left(\frac{2}{p-1}w+\xi {\partial }_{\xi }w\right)+{\partial }_{\xi }w.$$ 37

The unknown blow-up rate, $G\left(\tau \right):=\frac{{\partial }_{\tau }A}{A}$ is determined by imposing a template/pinning condition, e.g.,:

$$w\left(0,\tau \right)=1.$$ 38

To directly compute the self-similar blow-up solution, we solve the steady-state form of the MN system:

$$-\frac{{\mathrm{d}}^{3}w}{\mathrm{d}{\xi }^3}-\frac{\mathrm{d}{w}^p}{\mathrm{d}\xi }+G\left(\frac{2}{p-1}w+\xi \frac{\mathrm{d}w}{\mathrm{d}\xi }\right)+\frac{\mathrm{d}w}{\mathrm{d}\xi }=0.$$ 39

We consider a symmetric finite spatial domain ξ ∈ [−L, L], with L ≫ 1 and impose boundary conditions^70,71:

$$\begin{array}{ccc}\frac{2G}{p-1}w+\left(G\xi +1\right)\frac{\mathrm{d}w}{\mathrm{d}\xi }=0,\xi \hfill & =\hfill & \pm L\hfill \\ \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{\xi }^2}=0,\xi \hfill & =\hfill & L.\hfill \end{array}$$ 40

For the computation of the steady-state solution, we constructed a neural network ${\mathcal{N}}_w\left(\xi \right)$ with two hidden layers of 40 neurons each (tanh activation). The blow-up rate G is treated as a trainable parameter, eliminating the need for a separate network. The loss function includes PDE and boundary contributions, and training is performed using 2003 collocation points and the L-BFGS optimizer.

To enhance convergence, we initialized the network using smooth unimodal profiles (e.g., Gaussian distributions or the analytical traveling wave solution in Eq. 32) (see Section 4.2 in Supplementary Material for details). Importantly, regardless of the initialization, the network consistently converges to the same self-similar solution and blow-up rate. The resulting self-similar profiles are shown in Fig. 6 for representative values of p. PINN predictions are compared against solutions obtained using an in-house finite-volume solver⁷¹.

Fig. 6. Steady-state PINN predictions of self-similar blow-up solutions for the gKdV equation.

a Comparison of PINN (blue line with circles) and finite volume solutions (red dashed line) for p = 5.2. b Corresponding comparison for p = 5.6. In both cases, a strong agreement is observed in both the solution profiles and the inferred blow-up rates. The PINN uses 2003 collocation points, whereas the finite-volume solver employs 10,000 grid points.

To obtain solutions using the finite-volume discretization approach, we follow the methodology described in ref. ⁷¹. Specifically, we initialize the computation with an unstable traveling wave solution for p > 5 and introduce a small perturbation in the parameter, subsequently evolving the MN-dynamics until convergence to a stationary state. However, for this problem, the MN-dynamics exhibit significant convergence difficulties. In practice, convergence to a stationary profile is attained when the initial condition is chosen to be the analytical traveling wave solution (Eq. 32) and is supplemented with carefully tuned perturbations. Once convergence is obtained, we trace the branch of self-similar solutions using parametric continuation. The resulting finite-volume solutions are depicted in Fig. 6a, b for p = 5.2 and p = 5.6, with corresponding blow-up rates G = − 0.0430 and G = − 0.1083, respectively. In contrast, the PINN implementation converges robustly and without the need for specialized strategies. The predicted profiles show excellent agreement with finite-volume solutions, and the inferred blow-up rates are accurate. Specifically, after 15,000 training epochs (wall-clock time ~10 min with our hardware), the PINN predicts blow-up rates of G = − 0.0421 for p = 5.2 and of G = − 0.0967 for p = 5.6, in agreement with the corresponding finite-volume estimates, despite the fact that the finite-solver employs a considerably finer grid with 10,000 points, whereas the PINN uses 2003 collocation points.

Discussion

In this work, we developed a PINN-based framework for the solution of both the forward and inverse problems for nonlinear PDEs with continuous symmetries—specifically translations and scalings—by reformulating them as high-index DAEs in dynamically rescaled coordinates. These index-2 DAEs simultaneously encode the evolution of the invariant profile and the symmetry parameters (such as wave speed or similarity exponents). In particular, our MN-PINN framework enforces both the PDE residuals and the symmetry-reduction-induced algebraic constraints through the loss function.

The method was tested on canonical problems, involving traveling waves and first- and second-kind self-similar behavior (the latter presented in Supplementary Material, Section 2). In all cases, the proposed framework was able to solve the resulting constrained systems, making this a promising approach for studying scaling and front-propagating phenomena. One of the key advantages of this symmetry-reduced approach is its ability to handle problems that are otherwise difficult to solve using traditional numerical schemes. Many self-similar and moving-front problems would require adaptive meshes or moving domains to maintain resolution near sharp features. In contrast, the dynamic rescaling used here maintains the structure of steep or concentrating solutions (e.g., solitons or traveling waves) at a constant scale in the transformed coordinates, reducing the need for mesh refinement; notice that this is a potential advantage in comparison to PINN methods that operate in the original frame within such problems⁵⁸. This also tends to reduce stiffness and removes fast transients (we found that the reduced dynamics often converge to stationary or slowly varying states, facilitating training), potentially improving numerical stability. Additionally, our implementation bypasses the need to derive scaling conditions a priori, as well as the technical difficulty of large PINN residuals that can arise at under-constrained points when applying the method without pinning or template conditions⁶². In previous approaches, addressing this issue required introducing extra—often solution-specific—loss terms to enable identification of scaling exponents. In the present formulation, such considerations are systematically accounted for by the template-based MN formalism and are therefore absent.

Traditional solvers for PDEs (e.g., FD or FEM with method-of-lines) are typically computationally more efficient, but they struggle with high-index DAE formulations, requiring index reduction, projection methods or specialized time integrators that are not standard in environments like MATLAB and often demand advanced/specialized software that ensures consistent initialization and preserves algebraic and hidden constraints. On the other hand, PINNs offer a more flexible alternative: they unify differential and algebraic residual treatment within a single loss function, simplifying implementation when dealing with algebraic constraints and dynamic symmetries—not necessarily due to superior performance, but due to ease of use and adaptability. Moreover, the meshless and optimization-based nature of PINNs may provide advantages in higher-dimensional problems, where classical grid-based methods are strongly affected by the curse of dimensionality⁶⁷. Under suitable assumptions, neural-network approximations can exhibit approximation rates that are substantially less sensitive to dimension, such as Barron-type O(1/N) rates, compared to standard polynomial approximation methods⁷².

The approach is not without drawbacks. PINNs are known to be sensitive to hyperparameters and network architecture, and the choice of optimizer plays a crucial role. To this extent, we made an effort to systematically assess the robustness of our PINN framework to different random initializations and pretraining strategies in Supplementary Material Section 4, confirming that for the presented case studies, the method reliably converges to the correct self-similar profiles and scaling exponents across varying conditions. We used a simple feedforward network with tanh activations; other architectures may perform better or worse depending on the specific problem, but choosing optimal configurations is not trivial. We also observed that standard gradient-based optimizers, such as ADAM, often fail to achieve sufficient accuracy, particularly when the loss includes algebraic constraints that must be satisfied to high precision. The L-BFGS optimizer, although more effective in this context, is expensive. Its memory and computational cost scales poorly with network size, limiting the method to small- and medium-sized networks in practice. To mitigate convergence issues, we employed warm-up strategies, first fitting initial and boundary conditions before enforcing the full physics and algebraic constraints. We also tested greedy, continuation-inspired training schemes^52,53, where the solution is first learned on a small subdomain (e.g., close to initial conditions or pinning conditions) and then gradually extended. In the transformed coordinates, however, the dynamics were already sufficiently regular that such strategies offered no clear benefit - indicating that the symmetry reduction itself effectively regularizes the problem, removing the need for these additional training complexities. Additionally, in the present work, we have primarily focused on allowing the integrator to converge to a stable self-similar or traveling (or both) solution. However, as a proof of concept, we demonstrate the extension to stationary formulations, for the computation of self-similar blow-up solutions of the generalized KdV problem; a more comprehensive analysis of fixed-point approaches remains a topic of future work^66,73. Looking ahead, more efficient alternatives, such as parsimonious Random Projection Neural Networks (RPNN), as in ref. ³²—which were shown in ref. ³² to reach accuracies comparable to classical index-1 DAE solvers like ode15s and ode23t in MATLAB —might offer significant computational advantages, by employing randomized features and linear-algebraic solvers (e.g., SVD, COD) rather than gradient-based optimization. We plan to investigate RPNN-based PINNs with method-of-lines discretization to enable a more direct comparison with classical solvers while retaining the flexibility of the PINN framework. Moreover, it would be of particular interest to explore models with more complex dynamics and potential bifurcations of self-similar solutions, such as the nonlinear Schrödinger^13,74. In summary, this method provides a flexible approach to high-index DAEs arising from symmetry-driven PDEs—a class of problems where standard techniques struggle—opening a path toward structure-aware neural PDE solvers. However, systematic exploration of network architectures, hyperparameters, and optimization strategies remains necessary to identify best practices for the models of interest.

Methods

Problem statement

We consider the numerical solution of PDEs of the form

$${\partial }_{t}u\left(\mathit{x},t\right)={\mathcal{L}}_{\mathit{x}}\left(u\right),$$ 41

ideally posed on an unbounded spatial domain $\mathit{x}\in {\mathbb{R}}^d$, approximated numerically over a sufficiently large finite domain.

Our interest is in PDEs whose solutions exhibit symmetry-driven structures, such as self-similar profiles and traveling waves. These patterns often reflect scaling invariance, translational invariance, or other continuous symmetries inherent to the PDE dynamics.

Often, a self-similar solution to an autonomous evolution PDE, as in Eq. 41, in the variables (x, t) ≡ (x₁, x₂,…,x_d, t) takes the form

$$\mid t-t^{*}{\mid }^{\beta }F\left(\frac{x_1}{\mid t-t^{*}{\mid }^{{\alpha }_1}},\dots ,\frac{x_d}{\mid t-t^{*}{\mid }^{{\alpha }_d}}\right),$$ 42

where the similarity exponents α₁,…,α_d, β, and the function F are determined from the PDE and appropiate boundary and initial conditions. The time t^* is either a known initial time (thus setting t > t^*), or an unknown critical time (e.g., blow-up time, thus setting t < t^*). When the system also respects time-translation invariance, we can always assume, without loss of generality, t^* = 0.

To simplify the discussion, we begin by considering one-dimensional problems (d = 1). The PDE operator ${\mathcal{L}}_x$ is assumed to be, in general, nonlinear and to satisfy a scaling property. Specifically, for all A, B, C > 0, there exist constants $a,b\in \mathbb{R}$ such that

$${\mathcal{L}}_{\mathit{x}}\left(Bf\left(\frac{x-c}{A}\right)\right)=A^a{B}^b{\mathcal{L}}_y\left(f\left(y\right)\right),y=\frac{x-c}{A}.$$ 43

To capture symmetry-driven behaviors—such as co-traveling and self-similar dynamics-we consider a transformed ansatz w(y, τ) in rescaled space-time coordinates:

$$u\left(x,t\right)=B\left(\tau \left(t\right)\right)w\left(y\left(\tau \right)\equiv \frac{x-c\left(\tau \left(t\right)\right)}{A\left(\tau \left(t\right)\right)},\tau \left(t\right)\right),$$ 44

where A(τ) and B(τ) are unknown scaling functions, c(τ) is a time-dependent translation, and τ(t) is an internal rescaled time variable, for which no a-priori assumptions are made; the associated scaling is expected to be inferred from the PDE dynamics. The primary objective of this transformation is to reformulate the original PDE problem Eq. 41 into a form in which the dominant symmetry structures become stationary or slowly varying in time. Substituting Eq. 44 into the PDE in Eq. 43 yields the transformed equation

$$\begin{array}{c}\left({\partial }_{\tau }Bw+B{\partial }_{y}w\frac{-{\partial }_{\tau }c-{\partial }_{\tau }Ay}{A}+B{\partial }_{\tau }w\right){\partial }_t\tau =\hfill \\ =A^a{B}^b{\mathcal{L}}_y\left(f\left(y\right)\right).\hfill \end{array}$$ 45

Let us define the rescaled time τ(t) via

$${\partial }_t\tau =\sigma A^a{B}^{b-1},$$ 46

where σ = sgn(t − t^*). Then, substituting ∂_tτ as in Eq. 46 and dividing the transformed equation by B, we obtain

$$\sigma \left(\frac{{\partial }_{\tau }B}{B}w-\frac{{\partial }_{\tau }A}{A}y{\partial }_{y}w-\frac{{\partial }_{\tau }c}{A}{\partial }_{y}w+{\partial }_{\tau }w\right)={\mathcal{L}}_y\left(w\right).$$ 47

This transformation does not by itself yield a closed system (it is underdetermined): the profile w(y, τ) evolves according to the transformed PDE, whereas the scaling functions A(τ), B(τ), and c(τ) remain unspecified. To close the system, additional algebraic conditions are required. These may arise from normalization conditions, conservation laws, or template-based algebraic relations. These relations, combined with the transformed PDE, naturally lead to a system of DAEs. In many cases, the number of effective free parameters can be reduced via further problem-specific manipulations. A common approach is to introduce the blow-up/scaling rates G(τ) = ∂_τA/A of the solution width and C(τ) = ∂_τB/B of the solution amplitude, which allow reformulating the problem in terms of these logarithmic growth rates.

Several strategies exist to define such algebraic closures, each reflecting different physical or geometric properties of the evolving solution. In the following, we briefly outline common approaches and their mathematical characteristics.

Closure strategies for the transformed PDE system

In this section, we review several approaches to close the transformed PDE system in Eq. 47

(i) Pinning conditions. Pinning conditions fix geometric features of the solution profile w(y, τ) at a given spatial location y_* in the rescaled coordinates. Typical examples include fixing the value w(y_*) = w_*, a zero-crossing, or the location of a maximum. These constraints serve to remove translation and/or scaling ambiguities by anchoring the solution in space.

(ii) Conservation-based constraints. For PDEs with conserved quantities (e.g., mass, energy), these invariants can be enforced directly on w. Such constraints typically involve integrals of w, leading to integral algebraic relations, often of index-2. For example, if the total mass is conserved,

$${\int }_{-\infty }^{\infty }w\left(y,\tau \right)dy=K,$$ 48

where K is a constant.

(iii) Template conditions. To close the system in Eq. 47, we can also introduce a template function T, typically chosen based on prior knowledge, e.g., the initial condition or an approximate steady-state shape. A common strategy involves minimizing the L²-distance between the current profile w(x) and a scaled-translated version of T, thus defining the mismatch loss,

$$E\left(A,B,c\right):={\int }_{\mathrm{\Omega }}{\left(w\left(x\right)-B\cdot T\left(\frac{x-c}{A}\right)\right)}^{2}dx,$$ 49

and impose the following conditions, that correspond to minimization of Eq. 49

$${\frac{\partial E}{\partial A}\left.∣\right)}_{\begin{array}{c}A=1,\hfill \\ B=1,\hfill \\ c=0\hfill \end{array}}=0,{\frac{\partial E}{\partial B}\left.∣\right)}_{\begin{array}{c}A=1,\hfill \\ B=1,\hfill \\ c=0\hfill \end{array}}=0,{\frac{\partial E}{\partial c}\left.∣\right)}_{\begin{array}{c}A=1,\hfill \\ B=1,\hfill \\ c=0\hfill \end{array}}=0.$$ 50

Explicitly, the conditions in Eq. 50 correspond to the following algebraic constraints:

$$\begin{array}{cc}\hfill & {\int }_{\mathrm{\Omega }}\left(w-T\right)y{\partial }_{y}Tdy=0,\hfill \\ \hfill & {\int }_{\mathrm{\Omega }}\left(w-T\right)Tdy=0,\hfill \\ \hfill & {\int }_{\mathrm{\Omega }}\left(w-T\right){\partial }_{y}Tdy=0.\hfill \end{array}$$ 51

These provide three algebraic equations that constrain the instantaneous values of A(τ), B(τ), and c(τ), thus closing the transformed PDE system and yielding a DAE formulation. The extremization with respect to c provides information with respect to the wave speed, whereas the other two determine the width and amplitude information. In case these two are related in their evolution, the relevant conditions may degenerate into a single one for the “blowup/collapse rate”.

Related works: the method of slices, the freezing method, and MN-dynamics. Several techniques have been introduced to eliminate continuous symmetries, such as translations and scalings, in the dynamics of PDEs. One classical approach is the “method of slices,” introduced in ref. ⁷. This method defines a slice hyperplane that intersects the group orbit transversely, allowing the full system trajectory to be projected onto a reduced state space where the symmetry is factored out. The evolution in this reduced space is then coupled to reconstruction equations for the symmetry parameters. A related construction is the “MN-dynamics framework” in ref. ⁶, where self-similar solutions are studied by introducing moving and rescaled coordinates. The authors derive equations for both the evolving profile and the parameters (e.g., speed, scale) by imposing additional constraints-often involving moment conditions or orthogonality—that regularize the dynamics. Although similar in objective to the template-based method, MN-dynamics differs in the form of these constraints and in its treatment of the modulated profile. Such renormalization methods, already well-established in the dispersive wave literature, e.g., through the dynamic rescaling method and variants thereof^13,74, have recently been used to extract normal forms for the bifurcation of collapsing solutions^65,71, as well as to characterize the stability of the collapsing solutions (as stationary states) in the self-similar (so-called “co-exploding”) frame. The “freezing method,” proposed in ref. ⁷⁵, adopts a similar idea by transforming the PDE into a comoving and rescaled frame, yielding a differential-algebraic system. Phase conditions are imposed to “freeze” the solution in the transformed coordinates, typically by requiring orthogonality to the group tangent directions or by minimizing a suitable functional.

Index-reduction. One can arbitrarily select or combine any of the strategies described at points (i), (ii), or (iii) to impose a number of algebraic constraints matching the unknown transformation parameters (e.g., A(τ), B(τ), c(τ)). All the algebraic constraints presented can close the transformed PDE system, in Eq. 47, but result in an index-2 DAE, since they involve only linear functionals of w (either at specific spatial points or over domains) and do not explicitly depend on the transformation parameters. Consequently, two derivatives with respect to τ are needed to obtain an explicit ODE system. A common index-reduction technique involves differentiating the constraint once with respect to τ, reducing the system to index-1. Let the constraint take the generic form L_cons(w) + G_cons = 0, where L_cons is a linear operator, and G_cons is independent of w and τ. Differentiating yields

$${\partial }_{\tau }\left(L_{cons}\left(w\right)+G_{cons}\right)=L_{cons}\left({\partial }_{\tau }w\right)=0.$$ 52

Then, substituting the transformed PDE expression for ∂_τw, as in Eq. 47, the differentiated constraint, L_cons(∂_τw) = 0, becomes a relation involving explictly the unknowns A, B, c and their derivatives:

$$L_{cons}\left(\sigma {\mathcal{L}}_y\left(w\right)-\frac{{\partial }_{\tau }B}{B}w-\frac{{\partial }_{\tau }A}{A}y{\partial }_{y}w-\frac{{\partial }_{\tau }c}{A}{\partial }_{y}w\right)=0.$$ 53

This yields an index-1 DAE system that is more amenable to numerical integration.

In the context of PINNs, however, this index reduction is not strictly necessary. The original index-2 formulation can be enforced directly through the loss function, which simultaneously penalizes both the PDE residual and the algebraic constraints. This flexibility allows the method to bypass the derivation of auxiliary evolution equations, which are typically required by conventional solvers.

Numerical solution of DAEs with PINNs

In this section, and for the completeness of presentation, we first give a brief overview of the basic concept of physics-informed machine learning for the solution to differential equations in the form of PDEs/DAEs.

To approximate solutions of time-dependent PDEs involving evolving symmetry parameters, we consider a formulation combining differential and algebraic components. Let the main PDE be

$${\partial }_{\tau }w\left(y,\tau \right)={\mathcal{L}}_y\left(w;\mathit{p}\left(\tau \right)\right),y\in \mathrm{\Omega }\subset {\mathbb{R}}^d,$$ 54

with boundary conditions

$$\mathcal{B}w\left(y,\tau \right)=g\left(y,\tau \right),y\in \partial \mathrm{\Omega }.$$ 55

Here, $\mathit{p}\left(\tau \right)\in {\mathbb{R}}^m$ denotes a set of time-dependent parameters (e.g., scaling or translation functions, such as A(τ), B(τ), c(τ)), which appear explicitly in the transformed equation or influence the dynamics implicitly via the chosen frame.

To close the system, we assume that p(τ) must satisfy additional algebraic constraints:

$$f_j\left(w\left(\cdot ,\tau \right),\mathit{p}\left(\tau \right)\right)=0,j=1,\dots ,m,$$ 56

where each f_j may encode conservation laws, moment conditions, or template-matching criteria.

We introduce two neural networks: ${\mathcal{N}}_w\left(y,\tau \right)$ to approximate the solution w(y, τ), and ${\mathcal{N}}_{\mathit{p}}\left(\tau \right)$ to approximate the evolving parameters p(τ). Let ${\left\{y_i\right\}}_{i=1}^{n_y}\subset \mathrm{\Omega }$, ${\left\{y_{\ell }^{\partial }\right\}}_{\ell =1}^{n_{\partial }}\subset \partial \mathrm{\Omega }$, and ${\left\{{\tau }_k\right\}}_{k=1}^{n_{\tau }}$ be sampling points in space and time. The residual of the differential operator is incorporated into a loss function:

$${\mathcal{E}}_{\mathrm{pde}}:={\sum }_{i=1}^{n_y}{\sum }_{k=1}^{n_{\tau }}{∣{\partial }_{\tau }{\mathcal{N}}_w\left(y_i,{\tau }_k\right)-{\mathcal{L}}_y\left({\mathcal{N}}_w;{\mathcal{N}}_p\left({\tau }_k\right)\right)∣}^2,$$ 57

along with algebraic residuals

$${\mathcal{E}}_{\mathrm{alg}}:={\sum }_{j=1}^m{\sum }_{k=1}^{n_{\tau }}{∣f_j\left({\mathcal{N}}_w\left(\cdot ,{\tau }_k\right),{\mathcal{N}}_p\left({\tau }_k\right)\right)∣}^2,$$ 58

boundary residuals

$${\mathcal{E}}_{\mathrm{bc}}:={\sum }_{\ell =1}^{n_{\partial }}{\sum }_{k=1}^{n_{\tau }}{∣\mathcal{B}{\mathcal{N}}_w\left(y_{\ell }^{\partial },{\tau }_k\right)-g\left(y_{\ell }^{\partial },{\tau }_k\right)∣}^2.$$ 59

and initial condition residuals:

$${\mathcal{E}}_{\mathrm{ic}}:={\sum }_{i=1}^{n_y}{∣w\left(y_i,{\tau }_0\right)-w_0\left(y_i\right)∣}^2.$$ 60

The total loss is given by

$${\mathcal{E}}_{\mathrm{total}}={\lambda }_{\mathrm{pde}}{\mathcal{E}}_{\mathrm{pde}}+{\lambda }_{\mathrm{alg}}{\mathcal{E}}_{\mathrm{alg}}+{\lambda }_{\mathrm{bc}}{\mathcal{E}}_{\mathrm{bc}}+{\lambda }_{\mathrm{ic}}{\mathcal{E}}_{\mathrm{ic}}$$ 61

with weights λ_pde, λ_alg, λ_bc, λ_ic to balance contributions.

Backpropagation/ Automatic differentiation can provide the required time and spatial derivatives, as well as the gradients w.r.t trainable parameters of the PINN. This leads to a physics-informed neural approximation of a system of DAEs, where both the field variable and its symmetry parameters evolve consistently with the governing laws and closure conditions.

Inverse problems and data assimilation. The framework readily extends to inverse problems where some physical parameters are unknown or where sparse observational data are available. Let $\mathit{\theta }\in {\mathbb{R}}^q$ denote unknown parameters (e.g., diffusion coefficients, reaction rates, or nonlinearity exponents) to be inferred. We augment the loss function with a data mismatch term

$${\mathcal{E}}_{\mathrm{data}}:=\frac{1}{N_{\mathrm{data}}}{\sum }_{i=1}^{N_{\mathrm{data}}}{∣{\mathcal{N}}_w\left(y_i^{*},{\tau }_i^{*}\right)-u_{\mathrm{obs}}\left(x_i^{*},t_i^{*}\right)∣}^2,$$ 62

where $\left\{\left(x_i^{*},t_i^{*}\right)\right\}$ are sensor locations and observation times, u_obs are the corresponding measurements, and $\left(y_i^{*},{\tau }_i^{*}\right)$ are the transformed coordinates computed via the symmetry transformation (e.g., y = x − c(t) for traveling waves). The parameters θ are either incorporated as additional trainable variables in the network or treated as part of p(τ). The total loss becomes

$${\mathcal{E}}_{\text{total}}^{\text{inv}}={\mathcal{E}}_{\mathrm{total}}+{\lambda }_{\mathrm{data}}{\mathcal{E}}_{\mathrm{data}},$$ 63

enabling simultaneous inference of both the reduced dynamics and the unknown parameters from sparse, possibly offset measurements. This capability is particularly valuable when data are collected far from propagating fronts or when direct access to the symmetry-reduced field is unavailable, as demonstrated for the Nagumo PDE.

Ultimately, in the context of symmetry-reduced formulations—such as co-moving frames for traveling waves or self-similar scaling frames-additional algebraic constraints may arise from the coordinate transformation. These include template conditions, normalization constraints, or integral identities used to fix the degrees of freedom introduced by symmetry. PINNs can naturally incorporate these terms as soft penalties into the loss function, enabling the consistent solution of mixed PDE-DAE systems arising in the evolution of rescaled or traveling profiles.

Supplementary information

Supplementary Information ^{(2.6MB, pdf)}

Transparent Peer Review File ^{(186.4KB, pdf)}

Author contributions

M.E.K.: conceptualization, methodology, software, validation, formal analysis, investigations, computations, writing-original draft, writing-review and editing, and visualization. G.F.: methodology, software, validation, formal analysis, investigations, computations, data analysis, writing-original draft, writing-review and editing, and visualization. A.G.: data analysis, investigations, computations, writing-review and editing. C.S.: methodology, formal analysis, validation, writing-review and editing. P.G.K.: methodology, formal analysis, validation, writing-review and editing. I.G.K.: conceptualization, methodology, validation, resources, supervision, writing-review and editing, and visualization.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

G.F. and I.G.K. acknowledge the Department of Energy (DOE) support under Grant No. DE-SC0024162, as well as partial support by the National Science Foundation under Grants Nos. CPS2223987 and FDT2436738. C.S. acknowledges partial support by the PNRR MUR projects PE0000013-Future Artificial Intelligence Research-FAIR & CN0000013 CN HPC - National Centre for HPC, Big Data and Quantum Computing, Gruppo Nazionale Calcolo Scientifico-Istituto Nazionale di Alta Matematica (GNCS-INdAM).

Data availability

No external datasets were used in this study. All results and figures are generated from the governing PDE models within the simulation code. Warm-start and initialization files required for exact reproducibility of the reported results are included in the GitHub repository PINNs-for-DAEs.

Code availability

The code used to produce the findings of this study is publicly available on the GitHub repository PINNs-for-DAEs and archived with Zenodo as MN-PINN v1.0: PINNs for DAEs and Symmetry Driven PDE dynamics⁷⁶.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains Supplementary material available at https://doi.org/10.1038/s41467-026-75936-3.