Generative Neural Reparameterization for Differentiable PDE-constrained Optimization

TL;DR

This work applies the partial-differential-equation-constrained optimization technique to train a neural network that generates optimal parameters that minimize laser-plasma instabilities relevant to laser fusion and shows that the neural network generates many well performing and diverse minima.

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.

Figures (3)

• Figure 1: (a) The usual PDE constrained optimization problem where the parameters, $\theta_P$, are optimized to minimize the metric value, $M$. The required gradient is $\partial M / \partial {\theta_P}$. (b) Generative neural PDE constrained optimization comprises a neural network that maps a random variable to the optimal parameter set $\theta_P$. The relevant gradient is now $\partial M / \partial {\theta_N}$.
• Figure 2: Example spectra from a trained GNR model
• Figure 3: A comparison of the distribution of growth rates from simulations using the baseline uniform-random spectra and the spectra acquired using the GNR method shows a clear improvement over baseline.