Energy-Conserved Neural Pipelines: Attenuating Error Propagation in Modular Neural Networks via Physical Conservation Constraints

A diffusion model trained on Hamiltonian trajectories can achieve rollout MSE near $10^{-3}$, but the standard deviation of its energy over time is between 7500 and 36000 times larger than the ground-truth energy standard deviation, indicating a failure to preserve conservation laws. This gap motivates our central question of whether neural networks can learn or select globally conserved quantities from physical trajectories. We investigate this across three Hamiltonian systems: projectile motion, pendulum, and spring-mass. We use a structured $T(v)+V(q)$ energy model, a black-box Conservation Discovery Network (CDN), a polynomial CDN, and a conditional diffusion baseline. The structured network reaches $R^2 \geq 0.9999$ against analytical energy on clean data, while the black-box CDN reaches $R^2 \geq 0.996$ when trained with temporal consistency plus a small alignment loss to analytical energy at $t=0$ ($λ_{\mathrm{align}}=0.2$). With $λ_{\mathrm{align}}=0$, CDN Pearson $R^2$ collapses on pendulum and spring-mass ($< 10^{-3}$), showing that temporal consistency alone is not enough to reliably identify the true energy. Under $1\%$ additive Gaussian noise, the CDN outperforms the structured model on the projectile and spring-mass systems, suggesting that the CDN may be more robust to noisy inputs in this setting. However, the polynomial CDN is sensitive to training configuration: it achieves $R^2=0.78$ under a short training schedule on the pendulum system, but reaches $R^2=0.9998$ with more training time and data, regardless of whether noise is added.

Modern machine learning optimizes for accuracy without explicit treatment of internal computational cost, even though physical and biological systems operate under intrinsic energy constraints. We evaluate energy-aware learning across 2,203 experiments spanning vision, text, neuromorphic, and physiological datasets with 10 seeds per configuration and factorial statistical analysis. Three findings emerge. First, architecture alone explains negligible variance in accuracy (partial eta^2 = 0.001), while the architecture x dataset interaction is large (partial eta^2 = 0.44, p < 0.001), demonstrating that optimal architecture depends critically on task modality and rejecting the assumption of a universal best architecture. Second, a controlled lambda-sweep across lambda in {0, 1e-5, 1e-4, 1e-3, 1e-2} validates a single-parameter energy-regularized objective L = L_CE + lambda * E(theta, x): across this range, internal activation energy decreases by approximately three orders of magnitude relative to the unregularized lambda=0 baseline, with negligible accuracy change (<0.5 percentage points) on both MNIST and Fashion-MNIST. Third, energy-first architectures inspired by an action-principle framework yield 5-33% within-modality training-efficiency gains over conventional baselines. These results emerge from a research program that interprets learning through a structural correspondence between the action functional in classical mechanics, free energy in statistical physics, and KL-regularized objectives in variational inference. We frame this correspondence as a design hypothesis, not a derivation.

We introduce energy-based constraint networks -- a modality-agnostic architecture that learns structural coherence from contrastive pairs. The system processes frozen encoder embeddings through a state-space model with dual-head attention, producing a scalar energy measuring structural consistency alongside per-position energy scores that localize violations. Multiple independently trained branches detect different violation types and compose at inference without interference. We demonstrate the framework in two domains. In text, the system achieves 93.4% accuracy on trained corruption types and 87.2% on 9 unseen types, using frozen BERT and 7.4M trainable parameters. In vision, the same architecture achieves competitive deepfake detection: 0.959 AUC on FaceForensics++ Deepfakes and 0.870 on Celeb-DF without any Celeb-DF training data, using frozen DINOv2 and 3.6M parameters per branch. The framework supports flexible training: branches learn from designer-specified corruptions, real-world paired data, or both. Composable branches require representation compatibility -- a finding validated through extensive experimentation where five incompatible approaches failed before the compatible one succeeded. The architecture is encoder-agnostic and domain-agnostic: changing the domain requires only new corruption strategies; changing the encoder requires only a new input projection layer. To our knowledge, this is the first architecture to learn within-modality structural coherence as an explicit energy landscape with per-position decomposition, and to demonstrate that the same architecture transfers across modalities via corruption respecification alone.