Physics-Augmented Neural Networks

• Physics-Augmented Neural Networks are defined by embedding physical laws into model architectures to enforce properties like thermodynamic consistency, symmetry, and convexity by construction.
• They employ potential-based formulations, invariant-based inputs, and constrained architectures (e.g., ICNNs) to ensure scientifically admissible constitutive behavior.
• PANNs bridge data-driven methods and traditional physics models, enhancing trustworthiness and generalization in applications ranging from constitutive mechanics to particle physics.

Searching arXiv for recent and foundational papers on physics-augmented neural networks to ground the article. arxiv_search(query="all:\"physics-augmented neural networks\" OR all:\"physics-augmented learning\" OR all:\"input convex neural network\" constitutive", max_results=10) arxiv_search(query="physics-augmented neural networks", max_results=10) Physics-augmented neural networks (PANNs) are neural-network models in which physical structure is embedded into the architecture, constitutive representation, or training objective so that key scientific properties are satisfied by construction or enforced through explicitly structured auxiliary terms. In the contemporary literature, the term covers several related but non-identical practices: energy-based constitutive networks for finite-strain mechanics, generalized-standard-material formulations with learned free-energy and dissipation potentials, invariant-based models with structural tensors for anisotropy, reduced-order and hyperreduction surrogates that learn scalar energies rather than forces directly, and softer classifier formulations in which a neural network is trained jointly on prediction targets and physics-related auxiliary outputs (Liu et al., 2021, Klein et al., 2022).

1. Conceptual scope and relation to physics-informed learning

A central conceptual distinction in the literature is between physics-informed learning and physics-augmented learning. Physics-informed learning is formulated around a flexible model together with regularization or penalty terms that measure violation of a physical property, whereas physics-augmented learning uses a structured model decomposition in which one module satisfies the physical property by construction and a residual module captures deviations (Liu et al., 2021). This distinction is especially useful because many PANNs in mechanics are not trained by penalizing constitutive violations after the fact; rather, they restrict the admissible hypothesis class so that objectivity, symmetry, thermodynamic consistency, convexity, positivity, or monotonicity are inherited from the representation itself.

This architectural viewpoint also clarifies the relation between PANNs and PINNs. A study labeled as a “Physics-Informed Neural Network” in particle-physics model scanning, for example, uses a physics-informed layer, auxiliary physics outputs, and a combined classification-plus-physics loss, but does not enforce PDEs or boundary conditions; conceptually, it sits in the broader family of physics-augmented neural networks because the physics enters as a soft inductive bias through auxiliary observables rather than as a hard differential constraint (Vatellis, 2024). By contrast, many constitutive PANNs in solid mechanics are closer to architecture-level enforcement: they do not directly regress stress in a black-box manner, but learn a scalar potential and derive stress or forces by differentiation.

Across these strands, a recurring theme is that PANNs are intended to occupy a middle ground between purely data-driven neural surrogates and classical hand-crafted constitutive laws. This suggests that the defining feature of the field is not a single training recipe, but the use of physically meaningful latent structure—potential functions, invariant sets, internal variables, symmetry tensors, or constrained optimization formulations—to reduce the function class to scientifically admissible models.

2. Potential-based constitutive formulations

The most characteristic PANN formulation in mechanics is potential-based. Instead of predicting stresses, forces, or moments directly, the network represents a scalar energy-type quantity, and the observable constitutive response is obtained by differentiation. In finite electro-elasticity, for instance, the reversible internal energy density is written as

$$e = e(F,d_0), \qquad P(F,d_0)=\frac{\partial e(F,d_0)}{\partial F}, \qquad e_0(F,d_0)=\frac{\partial e(F,d_0)}{\partial d_0},$$

with the neural network parameterizing the internal energy in invariant form (Klein et al., 2022). In hyperelastic beam modeling, the effective beam strain-energy density $\psi(\epsilon,\kappa)$ is represented by a scalar-valued feed-forward neural network, and the stress resultants follow as

$$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$

which enforces a conservative hyperelastic closure at the beam level (Schommartz et al., 2024).

The same idea appears in reduced-order modeling. In non-intrusive hyperreduction, the network learns a reduced potential energy $\hat e(\mathbf{x}_r)$, and the reduced internal force and tangent stiffness are obtained as first and second derivatives:

$$\hat{\mathbf f}_r(\mathbf{x}_r)=\frac{\partial \text{PANN}(\mathbf{x}_r)}{\partial \mathbf{x}_r}, \qquad \hat{\mathbf K}_r(\mathbf{x}_r)=\frac{\partial^2 \text{PANN}(\mathbf{x}_r)}{\partial \mathbf{x}_r^2}.$$

This architecture guarantees symmetry of the tangent stiffness and positive semidefiniteness of the Hessian when the learned scalar is convex (Schütz et al., 15 Jan 2026).

Potential-based formulations are also the standard route to thermodynamic consistency in rate-dependent models. In small-strain viscoelasticity, the constitutive theory is organized around a free energy $\psi$ and a dissipation potential $\phi$, with stress and internal forces obtained from $\psi$ and the evolution of internal variables governed by the Biot relation

$$\frac{\partial \psi}{\partial q_\alpha} + \frac{\partial \phi}{\partial \dot q_\alpha}=0$$

(Rosenkranz et al., 2024). In finite-strain incompressible viscoelasticity, the same generalized-standard-material structure is combined with multiplicative decomposition of the deformation gradient and a dual dissipation potential (Kalina et al., 4 Nov 2025). In thermo-visco-plasticity, the free energy and dissipation potential are embedded in an input-convex, potential-based neural ordinary differential equation framework so that stress, internal-variable forces, flow rules, dissipation, and temperature evolution are all derived from learned thermodynamic potentials rather than regressed independently (Jones et al., 10 Dec 2025).

3. Physics embedded by architecture

PANNs typically embed physics through three coupled design choices: invariant-based inputs, constrained neural architectures, and analytical correction terms.

Invariant-based inputs are the standard device for objectivity and material symmetry. In isotropic finite-strain elasticity, the input set is commonly built from invariants of the right Cauchy–Green tensor $C=F^T F$ such as $\psi(\epsilon,\kappa)$0, $\psi(\epsilon,\kappa)$1, and $\psi(\epsilon,\kappa)$2 or $\psi(\epsilon,\kappa)$3 (Fuhg et al., 2023, Klein et al., 2023). For anisotropy, invariants are augmented by structural tensors. A particularly broad framework uses fully symmetric second-, fourth-, or sixth-order generalized structure tensors, enabling representation of isotropy, transverse isotropy, orthotropy, tetragonal, cubic, hexagonal, and more general anisotropy classes, while simultaneously calibrating the structure tensors during training so that the anisotropy type and orientation are inferred from data (Kalina et al., 2024).

The constrained architectures are usually ICNN variants. Input-convex neural networks are used when convexity is required in all network inputs; partially input-convex neural networks are used when convexity is required only with respect to deformation-related variables and not with respect to auxiliary parameters or design variables (Klein et al., 2023). In electro-elasticity, Softplus activations together with nonnegative weights make the learned internal energy convex in the chosen invariant coordinates, which in turn supports polyconvexity, thermodynamic consistency, objectivity, material symmetry, and appropriate volumetric behavior (Klein et al., 2022). In sparse model discovery, ICNNs are combined with positive, monotonically increasing networks so that hyperelastic energies, yield functions, and isotropic hardening laws each inherit the specific constitutive properties required by mechanics (Fuhg et al., 2023).

Analytical correction terms complete the construction. Zero-energy and zero-stress normalization at the reference state are often imposed by subtracting network values and linearized projections at the undeformed configuration (Schommartz et al., 2024, Kalina et al., 2024). Growth terms such as $\psi(\epsilon,\kappa)$4 or $\psi(\epsilon,\kappa)$5 are added to enforce coercivity or growth as $\psi(\epsilon,\kappa)$6 (Kalina et al., 2024, Klein et al., 2023). In nearly incompressible formulations, volumetric response is commonly separated analytically from the learned isochoric response, so that the network focuses on the nonlinear distortional part while pressure or penalty terms handle incompressibility (Klein et al., 2022, Jadoon et al., 7 Apr 2026).

4. Internal variables, dissipation, and path dependence

A large part of the current PANN literature extends beyond rate-independent hyperelasticity to path-dependent phenomena. Here the central question is not only how to represent a stored energy, but how to encode hidden state, dissipation, and evolution laws without surrendering thermodynamic structure.

In small-strain nonlinear viscoelasticity, the free energy is written as an equilibrium part plus a non-equilibrium part depending on an overstress-like strain $\psi(\epsilon,\kappa)$7, and the dissipation potential depends on $\psi(\epsilon,\kappa)$8 and $\psi(\epsilon,\kappa)$9. The distinctive training difficulty is that internal-variable labels are not available. One solution is to generate internal variables during training through a recurrent cell, particularly a long short-term memory cell, so that calibration requires only stress–strain paths and not prescribed internal variables (Rosenkranz et al., 2024). In finite-strain incompressible viscoelasticity, the inelastic deformation part is required to remain unimodular, and this is enforced by defining the dual dissipation potential in terms of a projected thermodynamic force; together with a trainable gate layer and $$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$0 regularization, the model can automatically identify the required number of Maxwell elements during training (Kalina et al., 4 Nov 2025).

Damage and phase-transformation variables are treated in a similar potential-based manner. For the Mullins effect, the damaged strain energy is written as

$$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$1

so that the damage mechanism scales an undamaged hyperelastic energy through a history-dependent scalar while preserving the energy-based constitutive structure (Zlatić et al., 2024). For strain-induced crystallization in natural rubber, a two-potential framework in the style of generalized standard materials uses neural-network free-energy and dissipation potentials together with two Lagrange multipliers and Karush-Kuhn-Tucker conditions to enforce the physically admissible bound $$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$2 for crystallinity (Friedrichs et al., 18 Nov 2025).

Thermo-visco-plastic PANNs push this logic further. They combine a learned free energy, a rectified convex dissipation potential, and a neural ODE for internal-variable evolution, while complementarity penalties are used to recover behavior ranging from finite elastic regions and KKT-like yield constraints to smooth rate-dependent and fully viscous inelasticity. The same framework is also used to predict the conversion of plastic work into heating from stress–temperature observations, rather than assuming a fixed Taylor–Quinney coefficient (Jones et al., 10 Dec 2025).

5. Numerical realization and solver integration

PANNs are not only training-time surrogates; a substantial part of the literature is devoted to making them usable inside established numerical solvers. One route is direct constitutive export. In explicit finite element analysis, pretrained hyperelastic PANNs can be trained in PyTorch, exported with architecture, weights, biases, activation choice, and material parameters, and automatically converted into standalone Fortran user materials for Simcenter Radioss and OpenRadioss. Runtime dependence on PyTorch is avoided; the generated routine receives the deformation gradient, evaluates invariants, performs a forward pass, backpropagates derivatives, and assembles the Cauchy stress natively inside the solver (Maurer et al., 29 Jun 2026).

Because constitutive evaluation occurs at every integration point and every time step, activation cost matters. In that implementation, replacing SoftPlus by SQuarePlus reduces isolated activation cost by about 87–88% faster after subtracting loop overhead. In the full finite element simulation, the element-force time increases only 3.6% over Carroll with SoftPlus and 1.8% with SQuarePlus for the one-hidden-layer PANN, and 15.6% with SoftPlus and 7.8% with SQuarePlus for the two-hidden-layer network, so the cheaper activation roughly halves the added runtime associated with the neural constitutive law (Maurer et al., 29 Jun 2026).

For implicit and dynamic simulations, tailored discretization has been developed as well. Hyperelastic PANNs have been combined with mixed Hu–Washizu-like formulations for nearly incompressible behavior and with an energy-momentum scheme based on discrete gradients, so that long-time dynamical simulations preserve energy and angular momentum while remaining compatible with the more intricate derivative structure of neural constitutive potentials (Franke et al., 2023). In beam modeling, the learned beam potential has been embedded into a mixed isogeometric collocation beam solver, demonstrating that the constitutive surrogate can operate inside an actual structural simulation rather than only in offline regression (Schommartz et al., 2024).

PANNs also appear as multiscale surrogates. In concurrent multiscale topology optimization, input-specific neural networks replace the microscale boundary value problem inside a finite-strain FE$$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$3-type loop by supplying homogenized free energy, stresses, and consistent tangent moduli from analytical first and second derivatives of the network, which makes macroscale optimization tractable for compressible, nearly incompressible, and anisotropic hyperelastic materials (Jadoon et al., 7 Apr 2026). In reduced-order modeling, PANN hyperreduction yields very accurate interpolation, but the same study reports quick divergence in extrapolation, especially under reversed loading, which prevents deployment in that benchmark and leads to a recommendation in favor of trajectory piecewise linear reduction when extrapolation robustness is critical (Schütz et al., 15 Jan 2026).

6. Domains of application

The application range of PANNs is already broad. In constitutive mechanics, they have been calibrated for isotropic and anisotropic hyperelasticity, nearly incompressible elasticity, finite electro-elasticity, incompressible viscoelasticity, Mullins-type damage, strain-induced crystallization, and thermo-visco-plasticity (Klein et al., 2022, Zlatić et al., 2024, Kalina et al., 4 Nov 2025, Jones et al., 10 Dec 2025). In electro-elasticity, an invariant-based convex neural network has been shown to handle analytical transversely isotropic data generated from an explicit potential, an analytically homogenized transversely isotropic rank-one laminate, and a numerically homogenized cubic metamaterial, with training datasets ranging from about 1.5 million points for the proof-of-concept study to about 1000 points and about 600 points for the homogenized examples (Klein et al., 2022).

Anisotropy identification and inverse design form another major branch. A generalized-structure-tensor PANN can infer anisotropy class and orientation from computational homogenization data while enforcing sparsity in the invariant set via a trainable gate layer and $$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$4 regularization (Kalina et al., 2024). Closely related inverse-design work uses partially input-convex neural networks to learn polyconvex free-energy surrogates from stress–strain data and then solve for microstructural design parameters and preferred directions using an evolution strategy, including cases where the test data have a different preferred direction from the one used in training (Jadoon et al., 2024).

Application breadth also extends beyond constitutive mechanics. In particle-physics model scanning, a PANN-like classifier with a dual-output structure predicts auxiliary physics quantities and then the viability label, using a total loss

$$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$5

with $$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$6, on an imbalanced dataset in which only about 16% of the points are viable (Vatellis, 2024). In atomistic modeling, PANNA provides a software pipeline for building neural interatomic potentials from physics-based local descriptors, with optional force training and export to molecular-dynamics packages such as LAMMPS and the KIM API (Lot et al., 2019). This suggests that the unifying theme of PANNs is not a single physical domain, but a common strategy of constraining neural representations by scientific structure.

7. Advantages, misconceptions, and limitations

The main advantage attributed to PANNs is that they improve trustworthiness and generalization by restricting the hypothesis class to functions that already respect part of the governing structure. In constitutive mechanics, this means that objectivity, material symmetry, thermodynamic consistency, convexity-related stability, normalization at the reference state, or monotonic dependence on selected parameters are embedded before training begins (Klein et al., 2022, Klein et al., 2023). In sparse model discovery, the same physics augmentation acts as a regularizer, and extreme sparsification with a smoothed version of $$n = \frac{\partial \psi}{\partial \epsilon}, \qquad m = \frac{\partial \psi}{\partial \kappa},$$7-regularization can reduce dense models with thousands of active weights to roughly 10–15 active parameters while preserving constitutive structure (Fuhg et al., 2023).

Several misconceptions recur in the literature. One is that a neural constitutive model is “physics-based” as soon as physics enters the loss. The surveyed work distinguishes sharply between soft penalties and architectural guarantees, and several papers explicitly position PANNs as alternatives to unconstrained stress regression or to PINN-style penalty-only formulations (Liu et al., 2021, Vatellis, 2024). A second misconception is that good interpolation and local mechanical consistency imply robust extrapolation. The hyperreduction study shows the opposite in a concrete benchmark: despite symmetry of the tangent stiffness, positive semidefinite Hessian, and zero-force consistency at the origin, PANN extrapolation can diverge quickly outside the training domain (Schütz et al., 15 Jan 2026).

The principal limitations are equally clear. Physics augmentation can reduce representational flexibility. In the Mullins-damage study, the most accurate model is the unconstrained one, while the fully polyconvex model is more restrictive and less accurate (Zlatić et al., 2024). Deeper or more strongly constrained networks also increase computational cost, even when efficient activations and native-code export are used (Maurer et al., 29 Jun 2026). Domain coverage remains incomplete: some implementations treat only nearly incompressible, isotropic, rate-independent hyperelasticity; some learn zero-reference conditions from data rather than imposing them hard; some exclude useful but non-polyconvex invariants; and some finite element workflows still lack consistent tangent operators for all solver classes (Klein et al., 2022, Maurer et al., 29 Jun 2026).

Taken together, the literature presents PANNs less as a single algorithm than as a constitutive principle for scientific machine learning: the neural network is permitted to approximate only within a physically curated function class, while the parts of the model that are already known—symmetry, convexity, dissipation, invariance, normalization, admissible bounds, or solver interface structure—are built in analytically. This suggests that the long-term significance of PANNs lies in their role as a bridge between mechanistic modeling and learned surrogates, rather than as a replacement for either.