20.6 Physics-Informed Learning

Core idea¶

Physics-Informed Learning must be treated as a system-level decision rather than an isolated technique. For a reproducible repository with versioned models, data, formulations, solvers, tests, results, and hardware evidence, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are interfaces, design and trajectory data, derivatives, metadata, seeds, tolerances, and provenance. The chapter-level formulation is

\mathrm{reproducibility}=\mathrm{model}+\mathrm{data}+\mathrm{configuration}+\mathrm{environment}+\mathrm{evidence}.

For this section, trace how the choice changes benchmarks and reports, the active constraints, and the implementable engineering design. A method is useful only when its assumptions are explicit and its result answers the same system question as the baseline.

Engineering interpretation¶

Ask three questions:

Which physical, informational, computational, or economic resource changed?
Which objective component or active constraint made the change valuable?
Does the conclusion survive model, disturbance, initialization, uncertainty, and implementation checks?

A practical action is to define benchmark. Record units and assumptions before optimization, report component objectives and margins afterward, and verify the result using an independent calculation or higher-fidelity model.

Activity 20.6: select a physics-informed surrogate¶

Activity 20.6: select a physics-informed surrogate

Four surrogates predict wave-energy-converter force. Validation uses data RMSE, normalized physics residual, energy-balance error, and inference time.

Physics weight $\lambda$	RMSE (kN)	Physics residual	Energy error	Time (ms)
0	3.8	0.120	9.0%	0.18
0.1	4.0	0.035	3.2%	0.20
1	4.7	0.009	1.1%	0.24
10	7.9	0.003	0.8%	0.25

Requirements are RMSE $\leq5$ kN, physics residual $\leq0.04$ , energy error $\leq4\%$ , and inference time $\leq0.30$ ms.

Determine feasible weights and compute normalized margins to every requirement.
For feasible models calculate $S=(\mathrm{RMSE}/5)^2+(r/0.04)^2+(e_E/0.04)^2+t/0.30$ and select the lowest score.
Under out-of-distribution waves, RMSE increases by factors $[2.1,1.6,1.3,1.2]$ and physics residual by $[2.4,1.7,1.4,1.3]$ . Reassess feasibility.
With five training seeds, the selected model has RMSE $[4.6,4.8,4.5,5.1,4.7]$ kN. Compute mean, sample standard deviation, and a 95% normal-approximation upper bound.
Decide whether physics-informed loss improves extrapolation enough for CCD use, separating evidence about prediction, physical consistency, and computational cost.