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

2026-03-19

AI Summary

This work proposes an end-to-end framework for solving fully nonlinear second-order partial differential equations—such as infinite-dimensional Hamilton–Jacobi–Bellman (HJB) and Kolmogorov equations—defined on separable Hilbert spaces, along with their associated optimal control problems, without resorting to finite-dimensional projections. The core approach, termed the Hilbert–Galerkin Neural Operator (HGNO), directly minimizes the L² norm of the PDE residual in the infinite-dimensional setting by integrating a deep Hilbert–Galerkin method with a Hilbert-space Actor-Critic reinforcement learning algorithm. Theoretically, the study establishes the first universal approximation theorem applicable to infinite-dimensional PDEs involving first- and second-order Fréchet derivatives and unbounded operators. Numerical experiments demonstrate the method’s efficacy and novelty by successfully solving deterministic and stochastic optimal control problems linked to the heat and Burgers equations.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

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• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 Fréchet 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 $L^2_μ(H)$-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.

Problem

• infinite-dimensional PDEs
• optimal control
• HJB equations
• Hilbert spaces
• nonlinear PDEs

Innovation

• Hilbert--Galerkin Neural Operators
• Universal Approximation Theorems
• Infinite-dimensional PDEs
• HJB equations
• Deep Hilbert--Galerkin Methods