Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data

TL;DR

This paper introduces a constrained optimization approach using neural networks to discover differential equations from data, improving robustness against noise and data scarcity compared to traditional unconstrained methods.

Contribution

It proposes a novel constrained optimization framework with intermediate state representation for neural network-based PDE discovery, outperforming penalty methods especially under noisy conditions.

Findings

Constrained methods outperform penalty methods in noisy data scenarios.

Neural network PDE discovery can be effectively combined with classical numerical solvers.

The approach enhances robustness and accuracy in data-driven PDE identification.

Abstract

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE by solving a constrained optimization problem and using an intermediate state representation similar to a Physics-Informed Neural Network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the PDE is satisfied at several spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on the Burgers' and the Korteweg-De Vreis equations demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. For both methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. We briefly highlight other small, yet crucial, implementation details.