Table 1.

Notations used throughout.

Variable	Definition	Domain
$K$	Pairwise interaction potential	$L_{\mathit{loc}}^1({\mathbb{R}}^d,\mathbb{R})$
$V$	Local potential	$C({\mathbb{R}}^d,\mathbb{R})$
$\sigma$	Diffusivity	$C({\mathbb{R}}^d,{\mathbb{R}}^{d\times d})$
$N$	Number of particles per experiment	{2, 3,…}
$d$	Dimension of latent space	$\mathbb{N}$
$T$	Final time	(0, ∞)
($\Omega ,\mathcal{B}$, $\mathbb{P}$, ${(\mathcal{F})}_{t\ge 0}$, ${(B_t^{(i)})}_{i=1}^N$)	Filtered probability space
${\mathbb{R}}^d$	Independent $\Omega ,\mathcal{B}$ Brownian motions on ($\mathbb{P}$, ${(\mathcal{F})}_{t\ge 0}$, ${(B_t^{(i)})}_{i=1}^N$, $X_t^{(i)}$)
$i\mathrm{th}$	$t$ particle in the particle system (1.1) at time ${\mathbb{R}}^d$	${\mathbf{X}}_t$
$N$	$t$-particle system (1.1) at time ${\mathbb{R}}^{Nd}$	${\mu }_t^N$
	Empirical measure of ${\mathbf{X}}_t$	$\mathcal{P}({\mathbb{R}}^d)$
$F_t^N$	Distribution of ${\mathbf{X}}_t$	$\mathcal{P}({\mathbb{R}}^{Nd})$
$X_t$	Mean-field process (3.2) at time $t$	${\mathbb{R}}^d$
${\mu }_t$	Distribution of $X_t$	$\mathcal{P}({\mathbb{R}}^d)$
$\mathbf{t}$	$L$ discrete timepoints	[0, $[0,T]$]
${\mathbb{X}}_{\mathbf{t}}$	Collection of $M$ independent samples of ${\mathbf{X}}_t$ at $\mathbf{t}$	${\mathbb{R}}^{MLNd}$
${\mathbf{Y}}_t$	Sample of ${\mathbf{X}}_{\mathbf{t}}$ corrupted with i.i.d. additive noise	${\mathbb{R}}^{MLNd}$
$U_t$	Approximate density from particle positions	$\mathcal{P}({\mathbb{R}}^d)$
$G$	Density kernel mapping ${\mu }_t^N$ to $U_t$	$L^1({\mathbb{R}}^d\times {\mathbb{R}}^d,\mathbb{R})$
$\mathcal{D}$	Spatial support of $U_t$, $t\in [0,T]$	Compact subset of ${\mathbb{R}}^d$
$\mathbf{C}$	Discretization of $\mathcal{D}$
${\mathbf{U}}_t$	Discrete approximate density $U_t(\mathbf{C})$
${\langle \cdot ,\cdot \rangle }_h$	semi-discrete inner product, trapezoidal rule over $\mathbf{C}$
${\langle \cdot ,\cdot \rangle }_{h,\Delta t}$	Fully-discrete inner product, trapezoidal rule over $\mathbf{C}\times \mathbf{t}$
${\mathbb{L}}_K$	Library of candidate interaction forces
${\mathbb{L}}_V$	Library of candidate local forces
${\mathbb{L}}_{\sigma }$	Library of candidate diffusivities
$\mathbb{L}$	(${\mathbb{L}}_K$, ${\mathbb{L}}_V$, ${\mathbb{L}}_{\sigma }$)
$\Psi$	Set of $n$ test functions ${({\psi }_k)}_{k=1}^n$	$C^2({\mathbb{R}}^d\times (0,T))$
${\varphi }_{m,p}(v;\Delta )$	Test functions used in this work (Eq. (4.4)
$\lambda$	Set of sparsity thresholds
$\mathcal{L}$	Loss function for sparsity thresholds (Eq. (4.6)