Physics-informed and mechanistic-informed learning
Mechanistic-informed learning incorporates known equations, constraints or scientific structure into a learning problem rather than relying only on observed input-output pairs.
Equation residual
If a system should satisfy
\[\frac{dy}{dt}=f(y,t),\]a learned approximation \(y_\theta(t)\) can be penalised when
\[R_\theta(t)=\frac{dy_\theta}{dt}-f(y_\theta,t)\]is far from zero.
A training objective may combine data error and mechanistic error:
\[L=L_{\mathrm{data}}+\lambda L_{\mathrm{mechanism}}.\]Biological constraints
Other constraints may enforce non-negative populations, conservation relationships or known parameter ranges.
Key idea. Mechanistic-informed learning uses scientific knowledge as part of the learning problem, not merely as an interpretation added after training.