AI RESEARCH
Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control
arXiv CS.LG
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ArXi:2603.19463v1 Announce Type: new 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.