Solving Implicit Inverse Problems with Homotopy-Based Regularization Path

TL;DR

This paper introduces a homotopy-based optimization approach for solving ill-posed implicit inverse problems, effectively handling nonlinearity, noise, and non-uniqueness by tracing a regularization path with efficient gradient computations.

Contribution

It presents a novel homotopy method combining regularization, adjoint-based gradients, and Newton-Raphson steps to improve stability and solution exploration in implicit inverse problems.

Findings

Method effectively traces solution paths with decreasing regularization.

Performance depends on ground truth sparsity and semi-convergence behavior.

Approach successfully applied to latent dynamics discovery in simulations.

Abstract

Implicit inverse problems, in which noisy observations of a physical quantity are used to infer a nonlinear functional applied to an associated function, are inherently ill posed and often exhibit non uniqueness of solutions. Such problems arise in a range of domains, including the identification of systems governed by Ordinary and Partial Differential Equations (ODEs/PDEs), optimal control, and data assimilation. Their solution is complicated by the nonlinear nature of the underlying constraints and the instability introduced by noise. In this paper, we propose a homotopy based optimization method for solving such problems. Beginning with a regularized constrained formulation that includes a sparsity promoting regularization term, we employ a gradient based algorithm in which gradients with respect to the model parameters are efficiently computed using the adjoint state method. Nonlinear constraints are handled through a Newton Raphson procedure. By solving a sequence of problems with decreasing regularization, we trace a solution path that improves stability and enables the exploration of multiple candidate solutions. The method is applied to the latent dynamics discovery problem in simulation, highlighting performance as a function of ground truth sparsity and semi convergence behavior.