# Optimization with learning-informed differential equation constraints and its applications

arXiv: https://arxiv.org/abs/2008.10893 · HTML: https://mlsanity.com/paper/2008.10893/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Aug 2020.

## Built on by (most cited first, 4 of 4)

- [CONFIDE: Contextual Finite Differences Modelling of PDEs](https://mlsanity.com/paper/2303.15827/) (2023)
- [On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs](https://mlsanity.com/paper/2306.05185/) (2023)
- [Mission-Aligned Learning-Informed Control of Autonomous Systems: Formulation and Foundations](https://mlsanity.com/paper/2507.04356/) (2025)
- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)

## Abstract (arXiv)

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.