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

arXiv: https://arxiv.org/abs/2603.19463 · HTML: https://mlsanity.com/paper/2603.19463/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 60 of its 128 references that are in the map.
- First posted on arXiv 19 Mar 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 30 of 60)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Learning in Mean Field Games: A Survey](https://mlsanity.com/paper/2205.12944/) (2022)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs](https://mlsanity.com/paper/2011.15110/) (2020)
- [Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks](https://mlsanity.com/paper/2102.11379/) (2021)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (2019)
- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024)
- [Mean-field neural networks: learning mappings on Wasserstein space](https://mlsanity.com/paper/2210.15179/) (2022)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)
- [Minimal Width for Universal Property of Deep RNN](https://mlsanity.com/paper/2211.13866/) (2022)
- [Signature Methods in Stochastic Portfolio Theory](https://mlsanity.com/paper/2310.02322/) (2023)
- [Extensions of the Deep Galerkin Method](https://mlsanity.com/paper/1912.01455/) (2019)
- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.06000/) (2023)
- [Accurate, scalable, and efficient Bayesian optimal experimental design with derivative-informed neural operators](https://mlsanity.com/paper/2312.14810/) (2023)
- [Neural Q-learning for solving PDEs](https://mlsanity.com/paper/2203.17128/) (2022)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization](https://mlsanity.com/paper/2512.14086/) (2025)

## Built on by (most cited first, 2 of 2)

- [Enforcing Dirichlet Boundary Conditions in Operator Learning](https://mlsanity.com/paper/2608.27256/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)

## Abstract (arXiv)

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 L2μ(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.