Title: Adaptive Fourier Decomposition-guided Neural Operator Design for Inverse PDE Problems
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Partial Differential Equation, Inverse Problems, Neural Operators, Banach Space, Adaptive Fourier Decomposition

Abstract: Inverse problems, which are generally ill-posed, aim to identify the unknown parameters of a physical system from the observations of its output. A large class of inverse problems for partial differential equations (PDEs) is only well-defined as mappings from operators to functions. However, existing operator learning frameworks either do not explicitly account for the underlying operator space or solve the inverse problems in a Hilbert space. In fact, it has been shown that a Banach space setting for the parameter space would be closer to reality for a wide range of problems. Motivated by this, we introduce AFDONet-inv, a novel neural operator solver whose design is rigorously guided by adaptive Fourier decomposition (AFD) theory, to solve inverse problems for PDEs in a Banach space. Each component of AFDONet-inv's architecture corresponds to an AFD operation in Banach space. Thus, AFDONet-inv is the first neural operator solver for inverse PDE problems whose architectural and component design of AFDONet-inv is fully guided by an established mathematical framework (in this case, AFD theory). This way, AFDONet-inv is mathematically explainable and grounded in the AFD theory and possesses several desirable properties. Extensive experiments demonstrate that AFDONet-inv outperforms state-of-the-art inverse PDE solvers in terms of solution accuracy.

TL;DR: We present a novel neural operator framework, whose design is mathematically grounded and explainable, to accurately solve inverse PDE problems.