Inverse problems for infinite-dimensional transport PDEs on Wasserstein space
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arXiv
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| Format: | Preprint |
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2025
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| author | Liu, Hongyu Qian, Jianliang Zhang, Shen |
| author_facet | Liu, Hongyu Qian, Jianliang Zhang, Shen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse problems for infinite-dimensional transport PDEs on Wasserstein space Liu, Hongyu Qian, Jianliang Zhang, Shen Optimization and Control Analysis of PDEs 35Q89, 35R30, 91A16, 49N80 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. |
| title | Inverse problems for infinite-dimensional transport PDEs on Wasserstein space |
| topic | Optimization and Control Analysis of PDEs 35Q89, 35R30, 91A16, 49N80 |
| url | https://arxiv.org/abs/2512.06871 |