Generative Neural Reparameterization for Differentiable PDE-constrained Optimization
This addresses the problem of finding multiple optimal solutions in PDE-constrained optimization for applications like laser fusion, though it is incremental as it builds on existing differentiable solver methods.
The paper tackles the limitation of PDE-constrained optimization in providing only a single optimal parameter set by reparameterizing parameters as neural network outputs to learn a distribution of optimal parameters, applying it to laser-plasma instabilities in laser fusion and showing generation of many diverse minima.
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.