Local Inverse Geometry Can Be Amortized
This lightning talk explores a breakthrough in solving nonlinear inverse problems for partial differential equations. The paper introduces Deceptron, a learned neural module that amortizes the expensive curvature computations traditionally required by classical optimization methods like Gauss-Newton. By training a reverse map to encode local inverse geometry and enforcing differential consistency through the Jacobian Composition Penalty, the resulting D-IPG solver achieves near-classical reliability while reducing solve time by orders of magnitude across seven PDE inverse benchmarks.Script
Every time you solve a nonlinear inverse problem for a partial differential equation, classical methods recompute expensive curvature information at every optimization step. This paper shows you can learn that geometry once and amortize it across thousands of solves, cutting solve time from seconds to milliseconds without sacrificing reliability.
The key innovation is Deceptron, a bidirectional neural module trained to satisfy local inverse consistency. The forward map approximates your PDE operator, while the reverse map learns to pull measurement-space corrections back into parameter space, mimicking the geometry of damped Gauss-Newton without solving any linear systems.
At each iteration, D-IPG forms a proposal in measurement space using the current residual, pulls it back through the learned reverse map, then applies relaxation and Armijo backtracking. The Jacobian Composition Penalty enforces that this learned pullback recovers first-order curvature geometry, with deviation bounded by local conditioning and composition error.
Across seven PDE inverse benchmarks including diffusion, reaction-diffusion, and Navier-Stokes systems, D-IPG achieves 94.8 percent mean success rate, converging in 5 to 7 iterations compared to over 30 for Gauss-Newton. On Allen-Cahn problems with basin structure, the Jacobian Composition Penalty increases basin occupancy from 18.75 percent to 82.5 percent, directly demonstrating its role in shaping convergence geometry.
Geometric analysis reveals that successful D-IPG updates span a spectrum from gradient-aligned to highly corrective, with Navier-Stokes achieving perfect reliability despite near-orthogonality to the gradient. This directly refutes the notion that the method simply learns rescaled gradient descent; it genuinely captures non-gradient curvature structure when local inverse geometry permits.
Amortizing local inverse geometry transforms the economics of repeated inverse solves, trading one-time training cost for orders-of-magnitude speedup across thousands of problems. To dive deeper into how learned operators can recover classical optimization geometry and explore more research at this intersection of numerics and machine learning, visit EmergentMind.com and create your own explainer videos.
