Physics-informed machine learning for reconstruction of dynamical systems with invariant measure score matching

TL;DR

This paper introduces PINN-IMSM, a mesh-free, physics-informed neural network framework that reconstructs high-dimensional dynamical systems from point-cloud data by leveraging invariant measure score matching and PDE-constrained optimization.

Contribution

The paper presents a novel mesh-free PINN framework using score matching for invariant measure reconstruction, scalable to higher dimensions and addressing ill-posed inverse problems.

Findings

Accurately recovers invariant measures of complex systems.

Successfully reconstructs dynamics in systems up to five dimensions.

Demonstrates robustness on chaotic and oscillatory systems.

Abstract

In this paper, we develop a novel mesh-free framework, termed physics-informed neural networks with invariant measure score matching (PINN-IMSM), for reconstructing dynamical systems from unlabeled point-cloud data that capture the system's invariant measure. The invariant density satisfies the steady-state Fokker-Planck (FP) equation. We reformulate this equation in terms of its score function (the gradient of the log-density), which is estimated directly from data via denoising score matching, thereby bypassing explicit density estimation. This learned score is then embedded into a physics-informed neural network (PINN) to reconstruct the drift velocity field under the resulting score-based FP equation. The mesh-free nature of PINNs allows the framework to scale to higher dimensions, avoiding the curse of dimensionality inherent in mesh-based methods. To address the ill-posedness of high-dimensional inverse problems, we recast the problem as a PDE-constrained optimization that seeks the minimal-energy velocity field. Under suitable conditions, we prove that this problem admits a unique solution that depends continuously on the score function. The constrained formulation is solved using a stochastic augmented Lagrangian method. Numerical experiments on representative dynamical systems, including the Van der Pol oscillator, an active swimmer in an anharmonic trap, and the chaotic Lorenz-63 and Lorenz-96 systems, demonstrate that PINN-IMSM accurately recovers invariant measures and reconstructs faithful dynamical behavior for problems in up to five dimensions.