Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs

TL;DR

Problem: propagate state uncertainty in SDEs via the FP-PDE and bound the PINN approximation error for the PDF $p(x,t)$. Approach: a recursive error-learning framework learns the error through a sequence of error functions $\hat e_i$ satisfying PDEs, with two PINNs guaranteeing arbitrarily tight bounds and a practical first-order bound using a single PINN that generalizes to other linear PDEs. Contributions: a theoretically grounded, efficiently computable error bound framework, feasibility analysis for the conditions, and empirical validation on nonlinear, high-dimensional, and chaotic systems demonstrating significant speedups over Monte Carlo. Significance: enables certified, scalable uncertainty propagation in complex SDEs for safety-critical applications.

Abstract

Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.

Figures (23)

• Figure 1: second-order error bound $B_2(t)$ on 1D Linear system
• Figure 2: $\alpha_1(t),\;t \in T'$ vs train loss of $\hat{e}_1$, for all first-order error bound experiments.
• Figure 3: First-order error bound results. (a)-(c) 1D Nonlinear PDF $p$ vs PINN $\hat{p}$, error $e_1$ vs error PINN $\hat{e}_1$, and error bound $B_1$ compared to the classical Gaussian mixture method $\hat{p}_{GM}$, illustrated at three time points. (d) 2D Duffing Oscillator true error $|e_1|$, error PINN $|\hat{e}_1|$, and error bound $B_1 \geq |e_1|$ over time. (e)-(f) 2D Inverted Pendulum PDF $p$, PINN $\hat{p}$, true error $|e_1|$, error PINN $|\hat{e}_1|$, and error bound $B_1 \geq |e_1|$ at $t=3$. (g)-(h) 3D Time-varying OU error $e_1$ and error PINN $\hat{e}_1$ at $t=1$.
• Figure 4: 1D Heat PINN solution $\hat{u}(x,t)$ and error $\hat{e}_1(x,t)$ v.s. true solution $u(x,t)$ and error $e_1(x,t)$.
• Figure 5: Training losses of $\hat{p}$ and $\hat{e}_1$

Theorems & Definitions (19)

• Remark 1
• Definition 1: $i$-th error and approximation
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1: Second-order error bound
• Theorem 2: Arbitrary tightness
• Remark 2
• Corollary 1: Space-time Error Bound
• Corollary 2: First-order error bound