On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs
This provides a rigorous foundation for training neural networks with nonsmooth activations in PDE-based learning problems, though it is incremental in extending low-regularity analysis.
The paper tackles the identification of a nonsmooth Nemytskii operator in semilinear elliptic PDEs to minimize solution error, proving that locally optimal operators are sigmoidal with a kink at zero and developing a convergent gradient projection algorithm.
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 $H^1_{loc}(\mathbb{R})$. 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.