Variational Inverse Discovery of Scalar Field Lagrangians: Automated Physics Law Recovery via Neural Networks

This paper introduces a variational framework for the automated discovery of physical laws from observational field data. By framing the discovery process as an inverse variational problem, we utilize Physics-Informed Neural Networks (PINNs) to recover the underlying Lagrangian density L(phi, d_mu phi) of a phi^4 scalar field. Unlike traditional methods that approximate solutions to differential equations, our LagrangianNet identifies the fundamental functional generator of the dynamics by minimizing the Euler-Lagrange residual. We demonstrate that the Principle of Least Action serves as a universal regularizer, allowing for the accurate reconstruction of non-linear potentials (parameters m^2 and lambda) without explicit supervision. Our method achieves convergence to an Euler-Lagrange residual of O(10^-7), successfully recovering the double-well potential characteristic of spontaneous symmetry breaking. This methodology represents a paradigm shift from data regression to functional discovery, enabling the extraction of fundamental laws directly from empirical observations.