Improved Dimensionality Reduction for Inverse Problems in Nuclear Fusion and High-Energy Astrophysics

TL;DR

The problem addressed is the need to solve inverse problems in nuclear fusion and high-energy astrophysics, which require exploring high-dimensional parameter spaces under substantial uncertainties. The authors propose a hybrid approach that combines Monte Carlo sampling with nonlinear dimensionality reduction and formal verification to enforce physics- and math-consistent parameter subspaces while accounting for uncertainties. They highlight that classical dimensionality reduction can produce invalid parameter combinations and present a path to certifiable subspaces by tagging disallowed regions and validating properties like $L^2$ stability, flux conservation, and thermodynamic consistency. If successful, this framework would enable efficient, trustworthy parameter estimation and design optimization for tokamak geometries and astrophysical inferences, improving fidelity without sacrificing physical validity.

Abstract

Many inverse problems in nuclear fusion and high-energy astrophysics research, such as the optimization of tokamak reactor geometries or the inference of black hole parameters from interferometric images, necessitate high-dimensional parameter scans and large ensembles of simulations to be performed. Such inverse problems typically involve large uncertainties, both in the measurement parameters being inverted and in the underlying physics models themselves. Monte Carlo sampling, when combined with modern non-linear dimensionality reduction techniques such as autoencoders and manifold learning, can be used to reduce the size of the parameter spaces considerably. However, there is no guarantee that the resulting combinations of parameters will be physically valid, or even mathematically consistent. In this position paper, we advocate adopting a hybrid approach that leverages our recent advances in the development of formal verification methods for numerical algorithms, with the goal of constructing parameter space restrictions with provable mathematical and physical correctness properties, whilst nevertheless respecting both experimental uncertainties and uncertainties in the underlying physical processes.