Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control

TL;DR

This paper introduces deep learning methods using Hilbert--Galerkin Neural Operators to approximate solutions of fully nonlinear second-order PDEs in infinite-dimensional spaces, with theoretical guarantees and practical algorithms demonstrated on control problems.

Contribution

It develops the first universal approximation theorems for such PDEs in Hilbert spaces and proposes novel deep learning algorithms that operate directly on infinite-dimensional spaces.

Findings

Proved UATs for functions and derivatives on Hilbert spaces.

Developed numerical methods that minimize residuals over the entire Hilbert space.

Successfully applied methods to control problems in physics and stochastic systems.

Abstract

We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fr\'echet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the $L^2_\mu(H)$-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.