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 phi^4 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…