2026 Theses Doctoral

Inverse Problems for Mean Field Games and Other Partial Differential Equation Models

Soedjak, Nathan

Inverse problems aim to infer unknown parameters in a mathematical model from partial measurements of the corresponding model solutions. This thesis is concerned with the theoretical and numerical analysis of a collection of partial differential equation (PDE) inverse problems, with applications spanning fields from economics to medical imaging.

The thesis is divided into two parts. In the first part, we study the problem of calibrating mean field game models, which provide tractable approximations of Nash equilibria for symmetric games with a large number of strategic players. Mean field games have emerged as a powerful framework for modeling crowd phenomena in economics, finance, traffic flow, and many other areas. From a mathematical perspective, these models are described by a coupled nonlinear PDE system consisting of a backward Hamilton–Jacobi–Bellman equation and a forward Fokker–Planck equation. Our contributions are as follows. We propose a novel and efficient algorithm for determining the potential cost function, inspired by the classical policy iteration method from optimal control theory. The numerical experiments are complemented by a rigorous convergence theory for the small time horizon regime. Additionally, we prove unique identification results for the interaction component of the cost functions. Some of our uniqueness results are particularly notable, as they are obtained in the multipopulation case and exhibit features unique to this setting.

In the second part, we pivot from inverse problems for mean field games to an assortment of other PDE inverse problems. The first of these inverse problems arises from the medical imaging technique of thermoacoustic tomography. The novelty in this work is that we use the second-order susceptibility as a source of contrast and must take into account the resulting second-harmonic generation nonlinear effect. Mathematically, the problem is to reconstruct the coefficients in a system of two nonlinear Helmholtz equations from certain measurements involving a combination of the coefficients and the amplitude of the (complex-valued) solutions throughout the domain; we prove uniqueness and stability theorems for these problems. The second inverse problem in this part is to reconstruct the initial condition of a Schrodinger equation from the amplitude of the solution at later times. For this phase retrieval problem in the Fresnel regime, we show that by utilizing amplitude data produced by multiple media, the initial wave field can be uniquely recovered.

In the final chapter, we propose an instance-wise adaptive sampling framework for constructing compact and informative training datasets for supervised learning of inverse problem solutions. We demonstrate through numerical experiments that this idea of dynamically allocating sampling effort based on the specific test instance enables significant gains in sample efficiency.