PIML (Physics Informed Machine Learning) integrates first-principles physics—typically expressed as ordinary or partial differential equations, conservation laws, symmetries, or constitutive relations—directly into the design, training loss, or architecture of ML (Machine Learning) models so that learned representations respect known physical structure. Practically this is realized through techniques such as physics-informed neural networks (PINNs) that add PDE residuals to the loss, hard-constraint parameterizations that enforce invariants, operator-learning approaches (e.g., DeepONets, Fourier Neural Operators) that learn mappings between function spaces, and hybrid schemes coupling differentiable solvers with data-driven components. The approach improves generalization and extrapolation in low-data regimes, enables simultaneous solution and parameter inference for forward and inverse problems, provides a route to uncertainty-aware scientific models when combined with Bayesian and ensemble methods, and facilitates scalable surrogate models for design, control, and real-time simulation. Theoretical concerns in PIML include ensuring stability and convergence when embedding continuous operators into discrete networks, identifiability of inferred parameters, and balancing data fidelity with physics constraints—issues addressed by tailored regularization, multi-fidelity training, adaptive weighting of constraint terms, and incorporation of inductive biases reflecting symmetries and conservation principles.
First formal uses of the "physics-informed" framing trace to mid-to-late 2010s (with prominent PINNs preprints and papers from around 2017–2019); the term and associated research area gained broad popularity and widespread adoption across computational science and engineering from roughly 2018 through the early 2020s as differentiable programming, scalable deep learning toolkits, and interest in scientific ML (Machine Learning) accelerated.