Partial Differential Equations and their Applications Seminar

Title: Spectral analysis of the material-independent modes for the Helmholtz equation

Abstract: In this talk I will discuss the spectral analysis of a family of nonself-adjoint spectral problems defined by a homogeneous Helmholtz equation when using the permittivity/permeability as the eigenvalue. The study of such problems is motivated by the modal decomposition approach in computational electromagnetism, wherein the electromagnetic field scattered by a nanocavity subject to radiation losses (i.e. in an open system), can be described as an infinite sum using as basis functions the eigenfunctions of a spectral problem defined by the unforced Maxwell's equations. Owing to radiation conditions, this linear spectral problem is non-self-adjoint, which is at the source of numerical and theoretical difficulties. Furthermore, the problem is non-standard in that its eigenvalues both diverge and accumulate at multiple finite points. I will present a rigorous spectral analysis of these modes and show their completeness in a suitable Hilbert space.

This is a joint work with Anne-Sophie Bonnet-Ben Dhia (ENSTA Paris) and Cristophe Hazard (ENSTA Paris).

Title: Global Convergence of Adjoint-Optimized Neural PDEs

Abstract:

Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks and solving the inverse problem of learning such terms from observed data in order to discover hidden physics. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to the available ground truth by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. These neural PDE models have emerged as an important research area in scientific machine learning. In this talk, we discuss the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity. Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove convergence of the trained neural-network PDE solution to the target data (i.e., a global minimizer). The global convergence proof poses a unique mathematical challenge that is not encountered in finite-dimensional neural network convergence analyses due to (i) the neural network training dynamics involving a non-local neural network kernel operator in the infinite-width hidden layer limit where the kernel lacks a spectral gap for its eigenvalues and (ii) the nonlinearity of the limit PDE system, which leads to a non-convex optimization problem, even in the infinite-width hidden layer limit (unlike in typical neural network training cases where the optimization problem becomes convex in the large neuron limit). The theoretical results are illustrated and empirically validated by numerical studies.

This talk is based on joint work with Justin Sirignano (University of Oxford) and Konstantinos Spiliopoulos (Boston University), see https://arxiv.org/abs/2506.13633 for the preprint.

TBA