Inverse problems aim to recover unknown signals from corrupted measurements, often under limited or unpaired data. In this talk, I will present two recent directions in Neural Optimal Transport motivated by inverse problems and function-space learning. First, I will introduce UOTIP, which formulates unpaired image inverse problems as an Unbalanced Optimal Transport map from noisy measurements to clean signals. By incorporating a likelihood-based cost, UOTIP admits a MAP-estimation interpretation, improves robustness to multi-level noise and class imbalance, and provides a theoretical guarantee for the existence and uniqueness of the transport map. Second, I will discuss HiSNOT, which extends Semi-dual Neural Optimal Transport to infinite-dimensional Hilbert spaces, providing a theoretical foundation for function-to-function transport maps relevant to Neural Operator learning. To address the spurious solution problem arising from the low-dimensional structure of functional data, HiSNOT employs principled Gaussian smoothing with provable convergence guarantees. Together, these works suggest a path toward extending Neural Optimal Transport from image inverse problems to stable operator learning in function spaces.

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(세미나 ZOOM 링크: https://cau.zoom.us/j/88050404196// 회의 ID: 880 5040 4196)

In this talk, we present a unified learning framework for inverse problems governed by wave and elliptic partial differential equations (PDEs), where the forward operator is unknown and no ground-truth interior data is available. The key idea is to embed a physics-based forward solver directly into the training loop, enabling learning from boundary measurement data alone. This removes the need for supervised training pairs and allows simultaneous recovery of unknown quantities. The framework is applied to three representative problems: (1) a nonlinear photoacoustic model where the sound speed depends on the unknown initial pressure, (2) a wave inverse problem with spatially varying unknown sound speed, connected to Calderón-type structures, (3) an elliptic inverse problem based on the Dirichlet-to-Neumann map, where theoretical uniqueness is available. Numerical results demonstrate robustness under noise. This work suggests a general paradigm for solving PDE inverse problems via physics-informed self-supervised learning.

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(세미나 ZOOM 링크: https://cau.zoom.us/j/88050404196 // 회의 ID: 880 5040 4196)