Physical Dynamics and Constraints › Why Physics Belongs Inside the Model
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In Part II, Chapter 6 we treated the transition model as a black box to be learned from data. That approach can work when the evaluation states resemble the training data and rollouts are short. Longer physical rollouts expose compounding rollout error: the model feeds its predictions back as inputs, so a small one-step bias can move the next state into a less-covered region and compound. Valid physical structure can rule out some impossible behavior in those regions, but it does not make the remaining predictions correct.
Physical structure attacks this problem from three directions:
- Inductive bias: Conservation laws and geometric structure restrict the hypothesis space. A hard parameterization or projection can remove entire classes of violating functions; a soft penalty merely makes them more costly during training.
- Extrapolation: Physical laws apply beyond the sampled states. Encoding a valid law can improve extrapolation and rule out some impossible predictions, but parameters and unmodeled effects still determine accuracy. A pendulum obeys the same nonlinear equation at every amplitude; representing that equation offers a better basis for extrapolation than memorizing observed amplitudes.
- Interpretability and debugging: An architecture that exposes energy, friction, or contact variables lets you compare them with measurements and constraints. This can narrow a diagnosis when the variables are observable and identifiable, though coupled mechanisms and numerical error may still produce the same symptom. Black-box models can be probed post hoc; structured models supply some diagnostics directly.
There is a tension to keep in view throughout. A fixed classical simulator enforces its encoded approximations but retains discretization error and model mismatch; calibration or learned components can improve it from data. A generic learned model can adapt to data without an explicit physical guarantee. Hybrid methods occupy the space between those endpoints, letting you decide which assumptions to enforce, which parameters to estimate, and which effects to leave flexible.