Inverse problems for infinite-dimensional transport PDEs on Wasserstein space

TL;DR

This paper develops a new theoretical framework for solving inverse problems related to infinite-dimensional transport PDEs on Wasserstein space, enabling the reconstruction of unknown components from observed data.

Contribution

It introduces a systematic approach using high-order calculus and variational schemes for inverse problems in nonlinear, nonlocal transport PDEs on Wasserstein space.

Findings

Unified method for inverse problems in mean field control and game equations

Reconstruction of cost functions and interaction kernels from value data

First comprehensive foundation for inverse problems in infinite-dimensional transport PDEs

Abstract

We develop a foundational framework for inverse problems governed by evolutionary partial differential equations (PDEs) on the Wasserstein space of probability measures. While the forward problems for such transport-type PDEs have been extensively and intensively studied, their corresponding inverse problems--which aim to reconstruct unknown operators, cost functions, or interaction kernels from observed solution data--remain largely unexplored at this level of generality. The cornerstone of our theory is a systematic approach featuring high-order calculus on the Wasserstein space and a progressive variational scheme. This methodology is specifically designed to address the challenges inherent in inverse problems for infinite-dimensional, nonlinear, and nonlocal transport PDEs. We demonstrate the power and versatility of our theory through two canonical examples: inverse problems for both the Mean Field Control (MFC) Dynamic Programming Equation and the Mean Field Game (MFG) Master Equation. Our work provides, for the first time, a unified foundation for identifying cost functions and interaction kernels from value function data. This establishes a new and fertile field of mathematical research with significant implications for both theory and applications in stochastic control and mean field games.