Deceptron: Learned Local Inverses for Fast and Stable Physics Inversion

TL;DR

This work tackles ill-conditioned inverse problems in physics and imaging by learning a local inverse of a differentiable forward surrogate via a bidirectional Deceptron, trained with a Jacobian Composition Penalty to align $J_g(f_W(x))J_f(x)$ with the identity. Inference uses the inverse-preconditioned gradient (D-IPG), which updates residuals in the output space and pulls back through the learned inverse under Armijo backtracking, effectively preconditioning inverse updates. Empirical results on Heat-1D and Damped Oscillator show substantial reductions in iteration counts relative to x-GD and competitive performance with Gauss-Newton, with RJCP serving as a diagnostic of conditioning; a single-scale DeceptronNet variant demonstrates fast, fixed-depth corrections in 2D and real-data Kodak24 experiments. A complementary focus on scalability and diagnostics suggests that Deceptron and DeceptronNet can enable faster, more reproducible scientific computing, with future work on multi-scale extensions and surrogate-range reporting.

Abstract

Inverse problems in the physical sciences are often ill-conditioned in input space, making progress step-size sensitive. We propose the Deceptron, a lightweight bidirectional module that learns a local inverse of a differentiable forward surrogate. Training combines a supervised fit, forward-reverse consistency, a lightweight spectral penalty, a soft bias tie, and a Jacobian Composition Penalty (JCP) that encourages $J_g(f(x))\,J_f(x)\!\approx\!I$ via JVP/VJP probes. At solve time, D-IPG (Deceptron Inverse-Preconditioned Gradient) takes a descent step in output space, pulls it back through $g$, and projects under the same backtracking and stopping rules as baselines. On Heat-1D initial-condition recovery and a Damped Oscillator inverse problem, D-IPG reaches a fixed normalized tolerance with $\sim$20$\times$ fewer iterations on Heat and $\sim$2-3$\times$ fewer on Oscillator than projected gradient, competitive in iterations and cost with Gauss-Newton. Diagnostics show JCP reduces a measured composition error and tracks iteration gains. We also preview a single-scale 2D instantiation, DeceptronNet (v0), that learns few-step corrections under a strict fairness protocol and exhibits notably fast convergence.

Figures (6)

• Figure 1: Deceptron: forward $f_W$ and reverse $g$ (instantiated as learned $g_V$ by default, or $g_{W^\top}$ if tied) with JCP; inference pulls output-space residuals back through $g$.
• Figure 2: Iteration distributions and convergence trajectories across problems. The right panel includes both Heat-1D and Oscillator RMSE curves (mean normalized RMSE).
• Figure 3: Scaling and ablations. D-IPG remains stable under increasing Heat-1D difficulty, JCP lowers composition error and iteration count, and final reconstructions confirm accuracy.
• Figure 4: DeceptronNet v0. A compact unrolled corrector using measurement and residual features.
• Figure 5: Scalability and diagnostics. DNet remains stable under harder real-data settings (Kodak24), while D-IPG shows decreasing $\mathsf{RJCP}$ throughout training, indicating improved local invertibility.