1. Nonlinear Inverse Problems and Scattering Theory
We obtained new results in the study of nonlinear wave equations on spacetimes that resemble Minkowski space at infinity. We showed that the nonlinear scattering data uniquely determine the topology, differentiable structure, and conformal geometry of the underlying spacetime, as well as the metric and nonlinearity up to a multiplicative factor.
In nonlinear optics, we demonstrated a new mechanism for non-detectability in external measurements. In media exhibiting second-harmonic generation, the induced higher-frequency field can remain entirely confined within a region, making the nonlinear interaction invisible to an external observer. This provides an example of invisibility arising not from geometric design but from nonlinear wave interaction.
2. Reconstruction from Noisy Data
A method was developed to reconstruct geometric objects (surfaces and submanifolds of high-dimensional spaces) from high-noise point clouds. A general approach was developed for interpolating smooth submanifolds from noisy data, including explicit error estimates. New geometric tools, such as the concept of R-exposedness, were introduced to characterize convexity properties relevant for stable reconstruction. The results show that, in sufficiently high-dimensional ambient spaces, submanifolds satisfying this condition are generic and can be recovered robustly from large, noisy data sets.
Closely related are advances in reconstructing Riemannian manifolds from partial measurements of the heat kernel. We established uniqueness and stability for this inverse problem even in families of manifolds with bounded curvature and diameter that may collapse to lower-dimensional stratified spaces. These results extend the geometric settings in which stable data-based reconstruction is possible.
3. Machine Learning for PDE-Based Inverse Problems
The project achieved substantial progress in combining analytic PDE theory with machine learning. For electrical impedance tomography (EIT), a classical imaging method, a new neural-operator method was developed. By extending the inverse map to a larger Hilbert space of kernel functions, machine learning can be applied in a larger space that contains typical measurement data. This enables efficient neural approximation.
Machine learning was also used to address stroke classification from EIT measurements. Using Virtual Hybrid Edge Detection (VHED) features as network inputs, higher classification accuracy was obtained than that achieved using traditional measurement data (the voltage-to-current map on the boundary of the body). Importantly, we observed that VHED features provide better performance under noise than raw data.
Furthermore, a new neural network architecture, Semialgebraic Neural Networks (SANNs), was introduced. SANNs can represent any bounded semialgebraic function. Such functions are defined by polynomial equalities and inequalities and can model discontinuities along smooth surfaces. This capability is essential for inverse problems involving sharp interfaces, such as those arising in imaging or material identification.
4. Novel Concepts and Methodological Innovations
A key conceptual development was the introduction of nonlinear scattering functionals, which allow the formulation of scattering problems even in regimes where classical scattering operators do not exist due to solution blow-up. This expands the range of nonlinear systems for which inverse scattering can be studied.
We also obtained a first solution to an inverse problem associated with variational inequalities, specifically the identification of an obstacle with Signorini-type boundary conditions. This motivates studies of a new class of nonlinear elliptic inverse problems that had not been previously explored.
