Generative Neural Reparameterization for Differentiable PDE-constrained Optimization

TL;DR

This paper introduces a neural reparameterization method for PDE-constrained optimization, enabling the generation of diverse optimal parameters rather than a single solution, useful in complex PDE landscapes.

Contribution

It proposes a neural network-based reparameterization approach for PDE-constrained optimization, allowing sampling of multiple optimal solutions in complex PDE problems.

Findings

Neural network can generate diverse optimal parameters.

Method effectively finds multiple well-performing minima.

Applicable to laser fusion optimization problems.

Abstract

Partial-differential-equation (PDE)-constrained optimization is a well-worn technique for acquiring optimal parameters of systems governed by PDEs. However, this approach is limited to providing a single set of optimal parameters per optimization. Given a differentiable PDE solver, if the free parameters are reparameterized as the output of a neural network, that neural network can be trained to learn a map from a probability distribution to the distribution of optimal parameters. This proves useful in the case where there are many well performing local minima for the PDE. We apply this technique to train a neural network that generates optimal parameters that minimize laser-plasma instabilities relevant to laser fusion and show that the neural network generates many well performing and diverse minima.