Local Inverse Geometry Can Be Amortized

TL;DR

This paper introduces a learned, amortized inverse geometry method called D-IPG that improves efficiency and accuracy in nonlinear inverse problems by using a surrogate Jacobian operator, outperforming traditional methods.

Contribution

The paper proposes a novel learned approach, D-IPG, that amortizes local inverse geometry using a surrogate Jacobian, achieving faster and more reliable solutions in PDE inverse problems.

Findings

D-IPG outperforms standard baselines on seven PDE inverse benchmarks.

Achieves 94.8% mean success rate across six reliability problems.

Reaches comparable or better recovery quality with up to 77x lower inference-time cost.

Abstract

Nonlinear inverse problems often trade inexpensive but fragile first-order updates against curvature-aware methods such as Gauss-Newton and Levenberg-Marquardt, which obtain stronger directions by repeatedly solving Jacobian-based linearized systems. We propose a learned alternative: amortize local inverse geometry into a reusable reverse operator. Our framework learns a bidirectional surrogate, Deceptron, and deploys it through D-IPG (Deceptron Inverse-Preconditioned Gradient), an iterative solver that pulls residual-corrected measurement-space proposals back to latent space. The key mechanism is a Jacobian Composition Penalty (JCP), which trains the reverse Jacobian to act as a local left inverse of the forward Jacobian; its runtime counterpart, RJCP, measures the same inverse-consistency error along optimization trajectories. We prove that D-IPG is first-order equivalent to damped Gauss-Newton under local pseudoinverse consistency, with deviation controlled by composition error and conditioning. Across seven PDE inverse-problem benchmarks, D-IPG outperforms standard baselines, achieves 94.8% mean success across the six-problem reliability suite, and reaches comparable or better recovery quality at up to 77x lower inference-time solve cost on the main benchmarks.