A Control Perspective on Training PINNs

TL;DR

This paper models the training of Physics-Informed Neural Networks (PINNs) as a stochastic control system and introduces adaptive controllers to improve convergence and robustness, providing a control-theoretic foundation for PINN training.

Contribution

It presents a novel control-theoretic framework for PINN training and proposes adaptive controllers to enhance convergence and robustness, a first step toward principled training algorithms.

Findings

Integral controller achieves accurate, robust convergence with correct physics models.

Leaky integrator improves performance under model mismatch.

Framework offers insights into PINN training dynamics and stability.

Abstract

We investigate the training of Physics-Informed Neural Networks (PINNs) from a control-theoretic perspective. Using gradient descent with resampling, we interpret the training dynamics as asymptotically equivalent to a stochastic control-affine system, where sampling effects act as process disturbances and measurement noise. Within this framework, we introduce two controllers for dynamically adapting the physics weight: an integral controller and a leaky integral controller. We theoretically analyze their asymptotic properties under the accuracy-robustness trade-off, and we evaluate them on a toy example. Numerical evidence suggests that the integral controller achieves accurate and robust convergence when the physical model is correct, whereas the leaky integrator provides improved performance in the presence of model mismatch. This work represents a first step toward convergence guarantees and principled training algorithms tailored to the distinct characteristics of PINN tasks.