Deducing Closed-Form Expressions for Bright-Solitons in Strongly Magnetized Plasmas with Physics Informed Symbolic Regression (PISR)

TL;DR

The paper tackles deriving analytical expressions for bright soliton solutions in strongly magnetized plasmas by solving the forward Maxwell–fluid reduced-order framework for the vector potential $A$ and density $n$ along the soliton coordinate. It introduces Physics-Informed Symbolic Regression (PISR), which couples symbolic regression with physics-based losses, boundary conditions, and data to discover closed-form expressions for the vector potential and number density. The results show that PISR can rediscover physically consistent soliton profiles that agree with prior numerical results, offering potential reductions in computation for reduced-order plasma models. Key methodological components include a fixed-depth prefix/postfix grammar, efficient symbolic differentiation, and a C++ implementation with LBFGS and Eigen, with guidance for further performance and robustness improvements.

Abstract

This paper presents a novel approach to finding analytical approximations for bright-soliton solutions in strongly magnetized plasmas. We leverage Physics-Informed Symbolic Regression (PISR) to discover closed-form expressions for the vector potential and number density profiles, governed by a reduced-order model derived from Maxwell-fluid equations. The PISR framework combines symbolic regression with physics-based constraints, boundary conditions, and available simulation data to guide the search for solutions. We demonstrate the effectiveness of the approach by rediscovering approximate solutions consistent with previously published numerical results, showcasing the potential of PISR for reducing simulation costs of reduced-order models in plasma physics.

Figures (2)

• Figure 1: Expression Tree Representations
• Figure 2: Comparison of Discovered Expressions and Numerical Simulation Data. $a(x) =\sinh\left(u(x)\right)-\alpha\cdot\gamma_{0}\cdot\tanh\left(u(x)\right)$