Empirical Model-Size Scaling for Neural PDE Solvers on the LQR-HJB Benchmark

Neural PDE solvers show empirical promise for high-dimensional partial differential equations, yet systematic studies of how approximation error scales with model size are lacking. We conduct a controlled empirical study comparing the Deep Galerkin Method (DGM) and Physics-Informed Neural Networks (PINNs) on the linear-quadratic regulator (LQR) Hamilton-Jacobi-Bellman equation — a benchmark admitting exact solutions at any dimension — in d in {2, 5, 10, 20} with both diagonal and coupled (non-separable) dynamics, measuring L2 relative error as a function of model size N in {10^3, 10^4, 10^5, 3x10^5}. We find that: (i) DGM exhibits approximate power-law scaling at d in {2, 5, 20}, though the trend is non-monotonic at d=10; (ii) PINNs scale comparably at d <= 5 but plateau at moderate error for d >= 10, with no improvement past N=10^4; (iii) PDE accuracy translates predictably to downstream control quality (R^2 = 0.91 in log-log space); (iv) these trends persist on coupled (non-diagonal) LQR, confirming they are not artifacts of dimension-separability.