# First-order conditions for the optimal control of learning-informed nonsmooth PDEs

arXiv: https://arxiv.org/abs/2206.00297 · HTML: https://mlsanity.com/paper/2206.00297/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2022.

## Built on by (most cited first, 3 of 3)

- [On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs](https://mlsanity.com/paper/2306.05185/) (2023)
- [C1-approximation with rational functions and rational neural networks](https://mlsanity.com/paper/2508.19672/) (2025)
- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)

## Abstract (arXiv)

In this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability.