Physics-Constrained Machine Learning

• Physics-Constrained Machine Learning is a hybrid modeling paradigm that integrates data-driven techniques with physical laws and invariants.
• It employs methods such as soft penalties, projection approaches, and simulator-in-loop training to ensure compliance with conservation laws and symmetries.
• PCML enhances predictive accuracy in diverse fields like turbulence modeling, environmental science, and inverse problems, despite optimization and scalability challenges.

Physics-Constrained Machine Learning (PCML) denotes a family of hybrid modeling strategies that combine machine learning with mechanistic or physical structure so that prediction is informed not only by data but also by conservation laws, symmetries, constitutive relations, simulator structure, or other scientific constraints. In the recent literature, closely related umbrella terms include “physics-informed machine learning” and “machine learning with physics knowledge,” and the same broad design space encompasses soft-constrained objectives, hard-constrained optimization, structured architectures, simulation-constrained learning, and data- or task-level transfer across related physical systems (Mukherjee et al., 28 Aug 2025, Nghiem et al., 2023, Watson et al., 2024).

1. Scope and terminology

PCML is used across a wider conceptual range than PDE-residual methods alone. In the dynamical-systems literature, physics-informed learning is defined to include not only physical laws but also abstract system properties such as symmetry, causality, conservation laws, stability, convexity, and invariance (Nghiem et al., 2023). Survey treatments further organize the field by where physics enters: objective functions, structured predictive models, data augmentation, and multi-task, meta, or contextual learning in which related datasets themselves act as carriers of physical regularity (Watson et al., 2024). A chemical-engineering perspective makes the taxonomy more explicit by distinguishing soft-constrained PCML, hard-constrained PCML, and general hybrid models (Mukherjee et al., 28 Aug 2025).

This broad scope implies that PCML addresses at least three distinct scientific tasks. One is solving a physical system when the governing equations are known. A second is inferring an unknown or partially known physical system from data. A third is generalizing across several instantiations of a physical system, as in operator learning, digital twins, or task families indexed by parameters, geometries, or boundary conditions (Watson et al., 2024). This suggests that PCML is best understood as a design philosophy rather than a single algorithmic family.

The terminology also encodes different strengths of enforcement. In some works, “physics-informed” refers primarily to soft residual penalties, whereas “physics-constrained” emphasizes exact or near-exact feasibility through constrained optimization, structural parameterization, or simulator-in-the-loop training (Valente et al., 11 Feb 2025, Karra et al., 2021). In practice, contemporary usage overlaps substantially, and many papers move along a continuum from weak inductive bias to hard feasibility.

2. Modes of imposing physics

The core mechanisms by which PCML imposes scientific structure can be summarized as follows.

Mechanism	Canonical form	Typical role
Soft-constrained learning	$$\min_\theta \lambda_d \mathcal{L}_d(\theta)+\lambda_p \mathcal{L}_p(\theta)$$	Encourages, but does not guarantee, physical consistency
PINN-style residual learning	Data, boundary, and PDE residual losses in one empirical risk	PDE solving and hybrid modeling
Output projection	$$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$	Enforces feasibility at prediction time
Simulator-in-the-loop training	$$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$	Uses an existing physics code as the governing constraint
Probabilistic constraints	Virtual observations such as $\mathcal{N}(0\mid c(X),\sigma_c^2 I)$	Embeds equality constraints into Bayesian inference
Physics-informed covariance	$y\sim \mathcal{GP}(f(x,\theta),k_\phi)$	Uses mechanistic mean structure and physics-aware discrepancy kernels

Soft-constrained formulations are the most common starting point. A general hybrid form is the penalized objective $$\min_\theta \lambda_d\,\mathcal{L}_d(\theta)+\lambda_p\,\mathcal{L}_p(\theta)$$, while PINN objectives add boundary and differential-operator residuals to supervised fit (Mukherjee et al., 28 Aug 2025, Doumèche, 11 Jul 2025). By contrast, projection-based physics-consistent learning trains any surrogate $y=f(x;\Theta)$ and then projects the output onto the manifold defined by equality constraints $g(x,p)=0$, so that feasibility is enforced on the final prediction rather than only encouraged during training (Valente et al., 11 Feb 2025). A different hard-constraint route is to embed a legacy simulator directly in the optimization loop and train the network subject to the discretized PDE solve, as in AdjointNet (Karra et al., 2021).

Probabilistic PCML replaces deterministic penalties by likelihood factors. In coarse-grained dynamics, equality constraints and residual equations can be introduced as “virtual observables,” for example $$p(\hat c_l=0\mid X_{l\Delta t},\sigma_c)=\mathcal{N}(0\mid c_l(X_{l\Delta t}),\sigma_c^2 I)$$, so that physical consistency becomes part of Bayesian state-space inference rather than a separate regularizer (Kaltenbach et al., 2019). A related gray-box Bayesian construction places the physics model in the mean function of a Gaussian process and uses covariance structure tied to parameter sensitivities and oscillatory discrepancy dynamics (Sedehi et al., 2023).

These mechanisms differ mainly in exactness, computational cost, and failure mode. Soft penalties are easy to integrate with standard optimizers but do not guarantee satisfaction on unseen inputs. Projection and simulator-constrained approaches can enforce admissibility up to solver tolerance, but they introduce nonlinear solves or adjoint computations into training or inference (Valente et al., 11 Feb 2025, Karra et al., 2021).

3. Structural and geometric constraints

A major branch of PCML constrains the hypothesis class itself. In “symmetry constrained machine learning,” the model is built on invariant features $\chi(x)$ satisfying $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$0 for all group elements $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$1, so the learned predictor is invariant by construction rather than only in expectation under data augmentation (Bergman, 2018). In the specific grayscale-inversion example analyzed there, the unconstrained architecture satisfies the structural bound $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$2 on accuracies over original and inverted data, which makes the cost of ignoring symmetry mathematically explicit (Bergman, 2018).

Monotonicity is another representation-level physical prior. A monotonic Gaussian process imposes derivative-sign information through a probit likelihood on virtual derivative observations. In the reported materials examples, the monotonic GP is strictly monotonic in the interpolation regime, yields a significant reduction in posterior variance, and is especially useful when data are scarce and noisy; in the extrapolation regime, however, the monotonic effect starts fading away as one goes beyond the training dataset (Tran et al., 2022).

Geometric mechanics supplies stronger architectural constraints. For separable Hamiltonian systems, Taylor-nets use symmetric Jacobian constructions and a fourth-order symplectic integrator so that the learned discrete flow preserves symplectic structure by construction. NSSNNs extend the same principle to nonseparable Hamiltonian systems through Tao’s phase-space augmentation and explicit symplectic splitting. RoeNet embeds the structure of Roe’s approximate Riemann solver for hyperbolic conservation laws, while NVM uses Helmholtz and Lagrangian vortex priors for incompressible flow (Tong, 2024). These methods are not residual-penalized black-box networks; they are structure-preserving numerical schemes with learnable components.

Hybrid models with residual components raise a separate structural issue: a sufficiently expressive black box can conceal a bad physical model. “Structural Constraints for Physics-augmented Learning” formulates two integrity criteria for this case: the residual model should be unable to replicate the physical model, and any best-fit hybrid model should have the same physical parameter as a best-fit standalone physics model (Kuang et al., 2024). This establishes a precise distinction between physically informed augmentation and what the paper calls “physics-misinformed machine learning” when the residual is allowed to absorb physical misspecification.

4. Optimization, simulation, and differentiable correction

PCML frequently turns learning into a constrained numerical optimization problem. In projection-based physics-consistent learning, the physical manifold correction is solved as a nonlinear program with CasADi Opti Stack and IPOPT. The method is applied after training, not as an implicit layer, and it guarantees exact constraint satisfaction only to the extent that the nonlinear solver converges (Valente et al., 11 Feb 2025). In sequential systems, the projected state can be recursively fed back into the model, so physical consistency also stabilizes trajectory rollout (Valente et al., 11 Feb 2025).

Another route is to differentiate through the corrective physics itself. In cloth modeling, a neural network predicts a mesh state that is then passed through either an inextensibility projection formulated as a second-order cone program or a quasistatic spring-energy relaxation. The training objective is placed on the corrected output, and gradients are propagated through the KKT system or equilibrium solve, so the network is optimized for the physically admissible result rather than for the raw pre-projection prediction (Geng et al., 2019).

Simulator-in-the-loop approaches retain the numerical properties of established codes. AdjointNet casts learning as PDE-constrained optimization in which the network proposes physical parameters, the embedded simulator solves $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$3, and adjoint or sensitivity information supplies $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$4 for gradient-based updates (Karra et al., 2021). In that formulation, physics is constrained everywhere in the domain, and properties such as consistency, stability, and convergence are inherited from the existing solver rather than approximated by autodiff re-discretization (Karra et al., 2021).

For stochastic deep-learning regimes, constrained optimization itself becomes a research topic. The Stochastic Penalty-Barrier Method (SPBM) addresses stochastic, non-convex, possibly non-smooth inequality-constrained learning with exponential dual averaging, stabilized penalty schedules, and a Moreau-envelope-inspired proximal step. The reported experiments show linear runtime overhead compared to unconstrained Adam for up to 10,000 constraints, but the paper explicitly states that it does not include theoretical convergence guarantees in the fully stochastic non-convex non-smooth setting (Bosák et al., 18 May 2026).

5. Application domains and empirical behavior

PCML has been demonstrated across environmental science, turbulence, surrogate modeling, inverse problems, and multiscale dynamics. In gapless land-surface-temperature mapping, a physics-constrained light gradient-boosting model augments remote-sensing predictors with Community Land Model forcing variables and CLM simulation outputs selected from surface-energy-balance reasoning. In sample-based validation it reports daytime RMSE $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$5 and nighttime RMSE $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$6; in space-based validation the corresponding RMSE values are $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$7 and $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$8 (Ma et al., 2023). The same study argues that physically selected forcing and state inputs improve extrapolation to extreme weather cases relative to purely empirical variants (Ma et al., 2023).

In turbulence closure for low-Prandtl-number fluids, the turbulent heat flux is represented through a tensor basis constrained by rotational invariance, dimensional analysis, and realizability, while an ANN predicts only the scalar coefficient functions from DNS-derived invariants. The model provides a complete vectorial representation of the turbulent heat flux and is trained and validated over $$\min_p \|p-f(x;\Theta)\|_W^2 \;\text{s.t.}\; g(x,p)=0$$9–$$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$0 (Fiore et al., 2022). This is representative of a recurrent PCML pattern: constrain the algebraic form first, then learn only the unresolved constitutive functions.

Projection-based surrogate correction has produced especially sharp empirical gains in moderate-dimensional systems. For a two-mass spring system, projection of neural-network outputs onto the energy-conserving manifold reduces energy conservation error from $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$1 for the plain neural network to $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$2 after projection, and reported rollout RMSE reductions reach 49.5% for $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$3 and 71.7% for $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$4 on a representative trajectory (Valente et al., 11 Feb 2025). In a low-temperature reactive plasma surrogate with 17 outputs, the same idea reduces RMSE of physical-law compliance by more than 9 orders of magnitude and particularly improves $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$5, $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$6, and $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$7 (Valente et al., 11 Feb 2025).

Probabilistic coarse-graining provides another application regime. By treating residual equations and conservation laws as virtual observations in a latent state-space model, the framework of (Kaltenbach et al., 2019) reconstructs high-dimensional particle and image-based systems while propagating uncertainty from latent states through coarse dynamics into fine-scale reconstructions. In the nonlinear pendulum example, the model produces accurate image forecasts out to 875 time steps, more than 11 times the training horizon (Kaltenbach et al., 2019).

Inverse parameter estimation with embedded simulators is a further recurring use case. In the lid-driven cavity-flow example of AdjointNet, the inferred viscosity is $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$8 against a ground truth of $$\min \mathcal{L}\;\text{s.t.}\;\mathcal{F}(\mathbf{u};\mathbf{p})=0$$9, corresponding to about $\mathcal{N}(0\mid c(X),\sigma_c^2 I)$0 relative error (Karra et al., 2021). In materials science, monotonic GPs show the same small-data advantage in a different form: on the fatigue-life example, the reported RMSE drops from $\mathcal{N}(0\mid c(X),\sigma_c^2 I)$1 for the regular GP to $\mathcal{N}(0\mid c(X),\sigma_c^2 I)$2 for the monotonic GP, while posterior variance is substantially reduced (Tran et al., 2022).

6. Limitations, controversies, and open problems

A central controversy in PCML concerns the status of the embedded physics itself. If the physics is wrong, then a hybrid model may become “physics-misinformed” rather than physics-informed, especially when a powerful residual model can conceal misspecification instead of exposing it (Kuang et al., 2024). This is why several recent works stress not only predictive accuracy but also structural integrity, identifiability, and decomposition non-overlap between the mechanistic and learned parts (Kuang et al., 2024).

The tradeoff between soft and hard enforcement remains unresolved. Soft residual penalties are flexible and easy to implement, but they do not guarantee exact feasibility on unseen inputs (Valente et al., 11 Feb 2025). Hard-constrained methods can enforce laws directly, yet they require explicit constraint functions, often need nonlinear optimization or simulator calls at inference or training time, and may become bottlenecks for high-dimensional outputs or expensive constraints (Valente et al., 11 Feb 2025, Karra et al., 2021). The chemical-engineering perspective identifies the unresolved design question directly: one must determine the amount and type of physical knowledge to embed, choose an effective fusion strategy with machine learning, scale models to large datasets and simulators, and quantify predictive uncertainty (Mukherjee et al., 28 Aug 2025).

Optimization difficulty is a separate issue. A recent survey notes that PINN-style methods remain hard to optimize, that balancing data and physics losses is delicate, and that cited benchmarking found finite-element methods often roughly 10 times faster and 100 times more accurate than PINNs on several test cases (Watson et al., 2024). This has shifted attention toward hybrid roles for PCML, such as warm starts, constrained surrogates, learned closures, and digital-twin augmentation rather than outright replacement of established numerical solvers (Watson et al., 2024).

Open problems are correspondingly broad. In control-oriented treatments, they include uncertainty quantification, minimal data requirements, convergence and hyperparameter issues, guarantees under noise and model mismatch, scalability of verification, and the computational burden of high-fidelity digital twins (Nghiem et al., 2023). In chemical engineering, they further include closed-loop experimental design, real-time dynamics and control, and multiscale phenomena (Mukherjee et al., 28 Aug 2025). Taken together, these results suggest that the long-term trajectory of PCML is less about a single canonical method than about principled ways of deciding which physical structures should be exact, which should be penalized, and which should remain learnable.