Neural Network Perturbation Theory (NNPT): Learning Residual Corrections from Exact Solutions

Many complex physical systems naturally decompose into an exactly solvable component augmented by a perturbative correction. Rather than directly employing neural networks to analyze complex physical systems, we introduce Neural Network Perturbation Theory (NNPT)--a correction learning approach that predicts residual perturbations after analytically subtracting known exact solutions. Using the gravitational three-body problem as testbed, we vary Jovian mass from f=0.05 to 30 times its physical value while holding network architecture fixed. An equalized-accuracy protocol with 1% tolerance reveals an unexpected non-monotonic capacity profile: capacity peaks at f=5 in the late integrable regime (3x32, 2242 parameters), remains elevated through the transition region (f~15-17), then decreases in the fully chaotic regime (f>=17, requiring only 2x32 with 1186 parameters)--a 47% reduction from peak. With symplectic integrator energy conservation below 2x10^{-4}, this counterintuitive phenomenon reflects genuine physical structure rather than numerical artifacts. Sequential correction experiments show negligible refinement (||y2||/||y1||~0.997), confirming single-stage networks capture dominant perturbative features without hierarchical decomposition. The capacity transition at f_c=16.6+-2.8 aligns with Chirikov's resonance-overlap criterion. Intermediate-complexity regimes impose maximal capacity requirements, while fully chaotic dynamics undergo ergodic smoothing--trajectory-specific fluctuations become irreducible noise, leaving only statistically smooth corrections requiring fewer parameters.