# On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs

arXiv: https://arxiv.org/abs/2306.05185 · HTML: https://mlsanity.com/paper/2306.05185/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 8 Jun 2023.

## Builds on (most cited first, 11 of 11)

- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [On Acceleration with Noise-Corrupted Gradients](https://mlsanity.com/paper/1805.12591/) (2018)
- [On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems](https://mlsanity.com/paper/2202.12262/) (2022)
- [Learning-informed parameter identification in nonlinear time-dependent PDEs](https://mlsanity.com/paper/2202.10915/) (2022)
- [A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives](https://mlsanity.com/paper/2210.07900/) (2022)
- [Optimization with learning-informed differential equation constraints and its applications](https://mlsanity.com/paper/2008.10893/) (2020)
- [First-order conditions for the optimal control of learning-informed nonsmooth PDEs](https://mlsanity.com/paper/2206.00297/) (2022)
- [Design of the monodomain model by artificial neural networks](https://mlsanity.com/paper/2107.03136/) (2021)
- [Discretization of parameter identification in PDEs using Neural Networks](https://mlsanity.com/paper/2108.10618/) (2021)
- [Optimal control of a non-smooth semilinear elliptic equation](https://mlsanity.com/paper/1705.00939/) (2017)
- [No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation](https://mlsanity.com/paper/2003.11478/) (2020)

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

- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)

## Abstract (arXiv)

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.