We present a neural force regression framework for discovering physical laws directly from noisy trajectory data. Unlike previous papers in this series, which minimized Euler-Lagrange residuals to learn a Lagrangian, this work identifies a fundamental gauge degeneracy in the separable Lagrangian architecture L = T(q̇) − V(q): the network can satisfy the Euler-Lagrange equations while keeping both T and V near-zero, making individual coefficient recovery impossible. We resolve this by training a neural network to directly fit observed accelerations q̈, then decomposing the learned force into conservative and dissipative components by velocity averaging. Integrating the conservative force recovers V(q) without gauge ambiguity. Applied to four mechanical systems — damped harmonic oscillator, nonlinear pendulum, coupled oscillator, and a sealed mystery signal — we recover the harmonic potential coefficient with 0.33% error and the damping coefficient with 0.007% error. The pendulum amplitude is recovered within 2.4%. The method is fully automated: data in, symbolic law out. This is Paper 6 of the Neural Lagrangian Series.
Muhammad Hanif (Mon,) studied this question.