Physics-Informed Machine Learning Overview

• Physics-informed machine learning is a paradigm that embeds governing equations, conservation laws, and structural constraints directly into ML models for enhanced physical plausibility.
• It includes diverse techniques such as PINNs, hybrid solver-in-the-loop systems, and physics-guided preprocessing applied in environmental, structural, and materials science.
• PIML improves model interpretability and generalization by effectively balancing soft and hard constraints, reducing computational costs while maintaining scientific consistency.

Physics-informed machine learning (PIML) denotes a family of methods that integrates prior physical information into machine learning models rather than treating prediction as a purely statistical input–output problem. In the surveyed literature, the physical prior may take the form of governing ODEs or PDEs, conservation laws, symmetries, constitutive relations, thermodynamic structure, or weaker structural constraints, and it may enter through the loss, the representation, the data pipeline, or a hybrid solver-in-the-loop architecture. Across these formulations, the stated goals are improved physical plausibility, data efficiency, interpretability, and generalization relative to purely data-driven models, while avoiding the rigidity or computational burden of fully hand-crafted simulators (Meng et al., 2022, Hao et al., 2022, Nghiem et al., 2023, Deshpande, 10 Nov 2025).

1. Conceptual scope and defining characteristics

PIML is not a single algorithm. The survey literature treats it as a modeling paradigm spanning neural simulation, neural operator learning, inverse problems, control, hybrid digital twins, and constrained forecasting, all unified by the explicit use of physical prior knowledge during learning (Hao et al., 2022, Nghiem et al., 2023). In this sense, PIML occupies the space between white-box or gray-box scientific models and unconstrained black-box ML.

A central boundary condition in the literature is that PIML is broader than physics-informed neural networks (PINNs). In some works, PIML is instantiated as a classical PINN that solves governing equations by minimizing PDE, boundary-condition, and initial-condition residuals with no labeled training data, as in the laser metal deposition thermal model where the unknown field is represented by a feedforward neural network $T_\theta(x,y,z,t)$ and trained against the heat equation and boundary flux conditions (Sharma et al., 2024). In other works, PIML is deliberately lighter-weight: the short-term flood model augments an LSTM with a “Trend Alignment” penalty and a temporal smoothness penalty rather than solving hydrodynamic PDEs (Gebre et al., 2 Jun 2026). A still different construction appears in synchrotron Laue microdiffraction, where the “physics-informed” component is an external filtering and pooling algorithm based on diffraction intensity and local spatial continuity, applied before a CNN autoencoder and clustering stage (Chan et al., 12 Jun 2025).

This diversity matters because one recurrent misconception is that “physics-informed” necessarily means PDE residuals inside a neural-network loss. The supplied literature explicitly rejects that equivalence. The flood-forecasting and grain-mapping studies argue that simple, high-level constraints or physics-guided preprocessing can already qualify as PIML when they encode reliable causal or measurement-generation structure (Gebre et al., 2 Jun 2026, Chan et al., 12 Jun 2025).

2. Modes of physical integration

The reviewed papers repeatedly classify PIML by where the physical prior enters: data, architecture, loss, or inference. The combustion review adds a second axis—soft versus hard constraints, and coordinate-to-variable versus field-to-field mappings—which sharpens this taxonomy for scientific ML (Wu et al., 3 Sep 2025).

Integration mode	Mechanism	Representative papers
Soft constraints in the loss	Penalize PDE residuals, conservation violations, trend inconsistency, or state-space inconsistency	(Sharma et al., 2024, Gebre et al., 2 Jun 2026, Bond et al., 2024)
Hard constraints in the representation	Encode invariance, exact BCs/ICs, stoichiometry, or structured dynamics directly in the model	(Meng et al., 2022, Hao et al., 2022, Wu et al., 3 Sep 2025)
Hybrid solver-in-the-loop models	Neural network predicts parameters, closures, or reduced representations for a physics solver	(Oddiraju et al., 2024, Nghiem et al., 2023, Deshpande, 10 Nov 2025)
Physics-guided preprocessing	Filtering, masking, pooling, or feature construction derived from measurement physics	(Chan et al., 12 Jun 2025)
Kernel or GP physical priors	PDE regularization induces an RKHS or GP model with physics-dependent covariance structure	(Doumèche et al., 2024, Daniels et al., 30 Oct 2025)

In soft-constraint formulations, the model remains free to violate physics, but such violations are penalized. The flood model is exemplary: the total loss adds a trend-alignment term enforcing non-negative discharge response after rainfall and a smoothness term penalizing unrealistic temporal oscillations (Gebre et al., 2 Jun 2026). The nonlinear steel frame metamodel uses an LSTM plus a physics residual derived from Newton’s second law and a state-space consistency term enforcing numerical differentiation constraints between displacement, velocity, and acceleration (Bond et al., 2024).

Hard constraints are stronger. Surveyed examples include Hamiltonian and Lagrangian neural networks, equivariant architectures, and chemistry-aware structures whose outputs obey algebraic invariants by construction (Meng et al., 2022, Hao et al., 2022). The combustion review describes hard constraints such as Softmax-based species parameterizations satisfying $\sum_k Y_k = 1$, analytically enforced BCs and ICs, and chemical-reaction-network architectures whose layers mirror mass-action kinetics (Wu et al., 3 Sep 2025).

Hybrid architectures place an explicit simulator inside the learning loop. The spacecraft thermal simulator is a clear instance: a neural network predicts reduced nodalizations under on-orbit thermal loads, and a coarse finite-difference thermal model then propagates the temperatures (Oddiraju et al., 2024). In biomolecular modeling, hybrid potentials take the form

$$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$

so that the ML component acts as a closure correction to a baseline physical energy landscape while preserving a mechanistic simulation backbone (Deshpande, 10 Nov 2025).

3. Mathematical formulations

A canonical formalization of PIML in the survey literature is empirical risk regularization by a physical inconsistency measure. In the kernel treatment of linear PDE priors, the objective is

$$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$

where $\mathscr{D}$ is a linear differential operator encoding the physical prior (Doumèche et al., 2024). This expression is mathematically close to the generic survey template

$$\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda_{\text{physics}}\mathcal{L}_{\text{physics}},$$

but it makes explicit that the physics term can be an operator norm rather than a pointwise residual (Hao et al., 2022).

In PINN-style coordinate-to-variable models, the learned field is a differentiable map $u_\theta(\mathbf{x},t)$, and automatic differentiation supplies $\partial_t u_\theta$, $\nabla u_\theta$, and higher derivatives inside residual losses. The laser metal deposition model uses precisely this construction, with PDE, initial-condition, and boundary-condition losses and no supervised temperature term in training (Sharma et al., 2024). The moving-load beam formulation does the same for Euler–Bernoulli dynamics, but replaces the singular Dirac source by a Gaussian approximation so that the physics residual becomes numerically tractable (Kapoor et al., 2023).

By contrast, high-level soft-constraint formulations may use physically motivated algebraic penalties rather than full operators. The flood LSTM defines

$$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{data}} + 0.1 \cdot \mathcal{L}_{\text{trend}} + 0.01 \cdot \mathcal{L}_{\text{smooth}},$$

where $\sum_k Y_k = 1$0 penalizes directional disagreement between rainfall and predicted discharge increments over a lag window $\sum_k Y_k = 1$1, and $\sum_k Y_k = 1$2 penalizes squared first differences of the hydrograph (Gebre et al., 2 Jun 2026). This is physically informed regularization in a deliberately non-PDE sense.

A further generalization appears in forecasting, where the “physical” or structural prior may be a linear constraint rather than a differential law. The dissertation on time-series PIML frames additive structure, break adaptation, forecast combination, transfer learning, and hierarchy reconciliation as constrained kernel problems of the form

$\sum_k Y_k = 1$3

thereby extending the PIML template beyond PDEs to weaker but operationally useful constraints (Doumèche, 11 Jul 2025).

4. Representative application patterns

In environmental prediction, PIML is repeatedly used as a robustness device under data scarcity and extrapolation. The short-term flood study augments a standard rainfall–runoff LSTM with trend-alignment and smoothness terms, and reports that under a 5% data regime the Nash–Sutcliffe Efficiency increases from $\sum_k Y_k = 1$4 to $\sum_k Y_k = 1$5, while a “2x precipitation” stress test shows that the baseline exhibits “saturation and erratic fluctuations” whereas the physics-informed model remains directionally consistent (Gebre et al., 2 Jun 2026). At monthly scale, the hydrological $\sum_k Y_k = 1$6-based PIML for the Narmada basin replaces selected empirical equations by ML submodels for evapotranspiration and streamflow, yielding test NSE values up to $\sum_k Y_k = 1$7 for $\sum_k Y_k = 1$8 and $\sum_k Y_k = 1$9 for $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$0, while annual water-balance deviation is reduced to $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$1 in the best mixed configuration (Barmparis et al., 2021).

In structural and mechanical dynamics, the physical prior often appears as governing equations or conservative structure. For moving loads on beams, approximating the Dirac delta with a Gaussian allows a PINN to recover forward deflections with relative errors $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$2, $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$3, and $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$4 for $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$5, and to infer load magnitudes as $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$6 and $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$7 in two inverse cases (Kapoor et al., 2023). For nonlinear steel moment-resisting frames, the PiML formulation combines model order reduction, wavelet analysis, LSTMs, and an equation-of-motion residual, and the reported behavior is uniformly better than a data-only LSTM across 1-, 5-, 9-, 14-, and 19-story cases, with performance saturating around 100 training simulations per building (Bond et al., 2024).

In manufacturing and materials characterization, PIML is used both as a PDE solver and as a physically guided inverse-analysis tool. The laser metal deposition study solves a 3D transient thermal problem with a data-free PINN and reports a global maximum absolute error of $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$8, corresponding to $$E_{\mathrm{hyb}}(\mathbf{x}) = E_{\mathrm{phys}}(\mathbf{x}) + E_\theta(\mathbf{x}),$$9 relative error against COMSOL, and $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$0 or $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$1 relative error in the top-surface melt-pool region (Sharma et al., 2024). The synchrotron Laue microdiffraction study combines a CNN autoencoder with an intensity-based, neighborhood-aware physics filter; the resulting PIML grain maps achieve an effective two-point-correlation FWHM of about $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$2, whereas both XMAS and ML-only maps remain at about $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$3, i.e. near the beam-size limit (Chan et al., 12 Jun 2025).

In thermal systems, hybrid PIML can act as a computational trade-off mechanism rather than a replacement for physical simulation. The spacecraft thermal simulator uses active nodalization: the neural transfer network predicts the coarse mesh, and a finite-difference solver propagates temperatures. The reported outcome is significantly better generalization than both a pure neural net and a fixed coarse-mesh model, while reducing computing cost by up to $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$4 relative to the high-fidelity model (Oddiraju et al., 2024).

The broader reviews show that these are not isolated designs. Combustion PIML is explicitly organized into chemical kinetics, reacting flows, and other combustion-related problems, with both coordinate-to-variable and field-to-field paradigms, and with soft and hard constraints spanning ODE kinetics, Navier–Stokes transport, stoichiometry, and flamelet tabulation (Wu et al., 3 Sep 2025). In biomolecular simulation, the same paradigm is framed as a response to the “biomolecular closure problem,” where differentiable simulation, hybrid physics–ML potentials, operator learning, and thermodynamic constraints are used to model long-timescale kinetics, rare events, and free-energy structure (Deshpande, 10 Nov 2025).

5. Theory, uncertainty, and diagnostics

One important development in recent PIML is the move from empirical success stories to statistical and operator-theoretic analysis. For linear differential priors, the kernel paper proves that the regularizer

$$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$5

defines an RKHS norm, so the PIML estimator is exactly kernel ridge regression with a physics-induced kernel $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$6 (Doumèche et al., 2024). This yields classical generalization machinery: the estimator converges at least at the Sobolev minimax rate $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$7, and in a one-dimensional example with $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$8 it approaches the near-parametric rate $$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s_{\mathrm{per}([-2L,2L]^d)}}^2 + \mu_n \|\mathscr{D}(f)\|_{L^2(\Omega)}^2,$$9 when the physical prior is exact (Doumèche et al., 2024).

The dissertation on mathematical foundations reaches a related but distinct conclusion for PINNs. It shows that naive discretized PINN training can overfit PDE residuals, then proves that adding ridge regularization restores risk consistency, and that for linear PDEs a Sobolev penalty yields strong convergence of the PINN estimator to the true solution while asymptotically satisfying the PDE and boundary conditions (Doumèche, 11 Jul 2025). This result sharpens a widespread empirical observation: low residual loss at collocation points is not, by itself, a reliable certificate of physical correctness.

The uncertainty-aware diagnostics paper addresses the practical model-selection problem that follows from multi-objective PIML. Its starting point is that simultaneous data and physics losses create ambiguity in model assessment and obscure epistemic uncertainty. Within a Gaussian-process PIML framework, it introduces the Physics-Informed Log Evidence (PILE) score as a single uncertainty-aware criterion derived from the marginal likelihood, and shows that minimizing PILE yields effective selections of kernel bandwidth, least-squares regularization, and kernel family; a “data-free” version of PILE can even identify kernels that are “well-adapted” to a given PDE before any measurements are collected (Daniels et al., 30 Oct 2025).

6. Limitations, misconceptions, and open directions

The literature is consistent that PIML improves physical plausibility more readily than it guarantees exact quantitative fidelity. In flood forecasting, the trend-alignment constraint suppresses unphysical fluctuations, but “capturing the exact magnitude of extreme peaks remains a challenge in sparse regimes” (Gebre et al., 2 Jun 2026). In moving-load mechanics, replacing the Dirac delta by a Gaussian is numerically effective, but it changes the original problem into one with a smeared load (Kapoor et al., 2023). In additive manufacturing, the PINN omits latent heat of fusion, melt-pool fluid flow, evaporation, and variable material properties (Sharma et al., 2024). In Laue grain mapping, the method depends on a hand-set $\mathscr{D}$0 intensity threshold and still “slightly overestimates grain sizes compared to EBSD” (Chan et al., 12 Jun 2025).

A second misconception is that adding any physics term automatically yields robust training. The survey literature instead emphasizes optimization pathologies, loss-balancing difficulties, benchmark scarcity, and architecture mismatch as persistent obstacles (Meng et al., 2022, Hao et al., 2022, Nghiem et al., 2023). The biomolecular review makes the same point in domain-specific language: model expressiveness, stability, stiffness, and nonequilibrium consistency remain unresolved, even in highly structured differentiable simulation pipelines (Deshpande, 10 Nov 2025).

The most recurrent forward directions are therefore not merely “more physics,” but better-calibrated integration. The supplied papers repeatedly call for transfer learning across regimes, adaptive or learnable physics constraints, stronger uncertainty quantification, differentiable physical learning stacks, digital twins, and diagnostics that unify data fit with physical consistency (Nghiem et al., 2023, Oddiraju et al., 2024, Deshpande, 10 Nov 2025). A plausible implication is that the mature form of PIML will remain heterogeneous: PDE-residual PINNs, kernelized physical priors, hybrid solver–ML systems, and physics-guided preprocessing will continue to coexist because they solve different parts of the same problem—how to constrain statistical learning by the structure of the physical world without forfeiting the flexibility that made machine learning attractive in the first place.