From Neural Lagrangians to Analytical Laws: Symbolic Extraction via Variational Distillation

Neural networks trained to discover physical laws typically function as black boxes, providing predictions without interpretable analytical forms. We present Variational Distillation, a novel framework that transforms trained Neural Lagrangians into human-readable symbolic equations. By combining Physics-Informed Neural Networks (PINNs) that minimize Euler-Lagrange residuals with symbolic regression (PySR), we extract analytical expressions for the Lagrangian density of a phi⁴ scalar field directly from trajectory data. Our pipeline achieves remarkable precision: the discovered potential energy formula matches the ground truth with a symbolic regression loss of 1. 531 × 10^-9, recovering the mass parameter (m²) with 0. 18% error, the coupling constant (lambda) with 1. 28% error, and the kinetic coefficient with 0. 18% error. This work establishes the first complete end-to-end pipeline for transforming neural black boxes into verified analytical physical laws, bridging the gap between machine learning and interpretable theoretical physics. The methodology is immediately applicable to discovering unknown governing equations in fluid dynamics, biophysics, and experimental physics. This is Paper 5 of the Neural Lagrangian Series, demonstrating that neural networks can be provably shown to encode correct physical theories by extracting and verifying analytical formulas.