Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control
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 ord...
| Authors | Samuel N. Cohen, Filippo de Feo, Jackson Hebner et al. |
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| Primary category | cs.LG |
| Published | 19 March 2026 |
| Last updated at source | 19 March 2026 |
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