Thermodynamic Structure-Informed Neural Networks

• Thermodynamic structure-informed neural networks are models that integrate physical constraints to produce evolution laws compliant with energy conservation and nondecreasing entropy.
• They employ reversible–irreversible decompositions and structured integrators to predict thermodynamic ingredients instead of directly fitting state transitions.
• These models are applied in reduced-order modeling, super-resolution, and constitutive learning, offering enhanced interpretability and robustness in dynamic systems.

Searching arXiv for recent and foundational papers on thermodynamic structure-informed neural networks and closely related formulations. arXiv search query: all:"thermodynamic structure-informed neural networks" OR all:"structure-preserving neural networks" OR all:GENERIC neural thermodynamics Thermodynamic Structure-Informed Neural Networks are neural architectures in which admissible predictions are restricted by thermodynamic structure rather than by an unconstrained state-transition map. In the foundational GENERIC-based formulation, the network is embedded inside a reversible–irreversible decomposition of dynamics, so that energy conservation and nondecreasing entropy arise by construction for isolated systems (Hernández et al., 2020). Across the subsequent literature, the same general idea appears in reduced-order modeling, super-resolution, constitutive learning, thermodynamic equations of state, and structure-aware PINNs: the learned object is not merely a trajectory fit, but a thermodynamically admissible evolution law, potential, or variational residual (Hernandez et al., 2020).

1. Definition and scope

In the canonical formulation, thermodynamic structure-informed neural networks do not learn dynamics “in an unconstrained, black-box way.” Instead, they are forced to generate time evolution through a thermodynamic formalism, most prominently the metriplectic structure of the General Equation for the Non-Equilibrium Reversible-Irreversible Coupling, GENERIC (Hernández et al., 2020). The central modeling move is to predict ingredients of a structured integrator—such as reversible and irreversible operators, or energy and entropy gradients—and then assemble the dynamics through an update rule whose admissible form is fixed by thermodynamics.

This distinguishes the approach from several nearby paradigms. It is not identical to generic physics-informed learning based on PDE residuals. The original GENERIC formulation explicitly emphasizes that no explicit balance equation or governing PDE needs to be written down and enforced directly; the prior is encoded at the level of geometry and thermodynamics rather than at the level of explicit differential equations (Hernández et al., 2020). In later PINN-based work, the phrase is broadened further: the residual itself may be replaced or supplemented by a thermodynamic formulation, such as Lagrangian mechanics, Hamiltonian mechanics, the Onsager variational principle, or entropy balance, rather than by a bare Newtonian equation (Li et al., 26 Mar 2026).

Related terminology reflects this broader scope. “Structure Preserving Neural Networks” are neural networks whose outputs parameterize a thermodynamically admissible evolution law rather than the next state directly (Urdeitx et al., 2024). “THINNs: Thermodynamically Informed Neural Networks” adopt a different but conceptually allied strategy: they choose the penalty in PINNs from the large-deviation rate functional of the underlying fluctuating system, so that the loss penalizes improbable deviations rather than a heuristic norm of the residual (Castro et al., 23 Sep 2025). This suggests that the field is best understood as a family of inductive-bias strategies grounded in thermodynamic admissibility rather than as a single architecture.

2. GENERIC and the reversible–irreversible split

The most influential formal backbone is the GENERIC decomposition

$$\dot z = L(z)\,\frac{\partial E}{\partial z} + M(z)\,\frac{\partial S}{\partial z},$$

where $E(z)$ is the energy potential, $S(z)$ is the entropy potential, $L$ is a skew-symmetric Poisson operator, and $M$ is a symmetric positive semidefinite friction operator (Hernández et al., 2020). The first term is the reversible, Hamiltonian-like contribution; the second is the irreversible dissipative contribution.

GENERIC imposes the degeneracy conditions

$$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$

These conditions enforce the separation between reversible and irreversible physics: the reversible part does not produce entropy, and the dissipative part does not change energy. Consequently,

$\dot E = 0,\qquad \dot S \ge 0,$

so the first and second laws are encoded structurally rather than recovered post hoc (Hernández et al., 2020).

In the discrete-time implementation of the original method, the evolution is approximated with a forward Euler step,

$$z_{n+1}=z_n+\Delta t\left(\mathsf L\,A(z_n)+\mathsf M\,B(z_n)\right),$$

where $A(z)\approx \partial E/\partial z$ and $B(z)\approx \partial S/\partial z$ are discrete gradients. The discrete degeneracy constraints

$E(z)$0

are added as a penalty in training (Hernández et al., 2020).

A closely related comparison is between the double-generator GENERIC formalism and a single-generator formalism based on a generalized free energy $E(z)$1. In the single-generator case,

$E(z)$2

whereas GENERIC keeps the reversible and irreversible generators distinct through $E(z)$3 and $E(z)$4. The comparative literature characterizes the former as less constrained and more expressive, and the latter as more explicit about thermodynamic admissibility because it enforces the degeneracy conditions directly (Urdeitx et al., 2024).

3. Neural architectures and training objectives

The original structure-preserving network is a feedforward neural network that maps the current state $E(z)$5 to structured outputs $E(z)$6 and $E(z)$7, rather than to $E(z)$8 directly. The fixed parts are the integration rule, the thermodynamic constraints, and usually the Poisson and friction matrices $E(z)$9. The output dimension scales like $S(z)$0 for an $S(z)$1-dimensional state. Training combines a one-step data term, a degeneracy penalty, and regularization, so the data determine numerical values while the structure determines admissible forms and couplings (Hernández et al., 2020).

In reduced-order settings, the same principle is transferred to latent variables. A sparse autoencoder first identifies a low-dimensional latent manifold, and a second network then learns latent dynamics while enforcing GENERIC. In this formulation, the structure-preserving neural network outputs

$S(z)$2

and advances the latent state through

$S(z)$3

Here the architecture is designed so that $S(z)$4 is skew-symmetric and $S(z)$5 is symmetric positive semidefinite; for $S(z)$6, positive semidefiniteness is enforced through a Cholesky-type construction (Hernandez et al., 2020).

The super-resolution variant replaces the sparse autoencoder with an adversarial autoencoder. Low-resolution fields are encoded into a latent vector, the decoder reconstructs both the original low-resolution field and a higher-resolution field, and a latent SPNN then advances the code in time through the same GENERIC update. In this version, the structural constraints are enforced by reshaping the outputs as

$S(z)$7

which guarantees skew-symmetry and positive semidefiniteness at the algebraic level (Bermejo-Barbanoj et al., 2024).

These implementations share a common design principle. The network does not directly predict the next state; it predicts thermodynamic ingredients—potentials, gradients, or operators—from which a time integrator or constitutive response is assembled. Thermodynamic structure is therefore encoded both at the architecture level and at the loss level (Urdeitx et al., 2024).

4. Extensions beyond direct state evolution

The same structure-informed idea has been extended from dynamical-system learning to reduced-order modeling, super-resolution, constitutive modeling, and equations of state. In each case, the neural network learns a thermodynamic object whose derivatives or induced geometry determine admissible observables.

Direction	Structured object learned	Representative paper
Reduced-order dynamics	Latent GENERIC evolution law	(Hernandez et al., 2020)
Super-resolution forecasting	AAE latent manifold + SPNN dynamics	(Bermejo-Barbanoj et al., 2024)
Thermoelastic constitutive learning	Helmholtz free energy $S(z)$8	(Fuhg et al., 2024)
Multiphase EoS learning	Potential $S(z)$9 or symplectomorphism	(Kevrekidis et al., 2024)
Coupled thermomechanics	Internal energy $L$0 and dissipation potential $L$1	(Holthusen et al., 30 Mar 2026)

In thermoelastic constitutive modeling, the learned quantity is not stress directly but a thermodynamically constrained Helmholtz free-energy density. For stable thermoelasticity, the potential is required to be polyconvex in deformation and concave in temperature. The paper denotes such a free energy as polyconvex-concave (PCC), and uses input-convex neural networks for deformation-dependent terms together with temperature-dependent factors chosen so that the overall free energy remains PCC (Fuhg et al., 2024). Stresses and entropy are then obtained by differentiation of a single potential.

A distinct route replaces the Helmholtz description by internal energy and a dissipation potential. In that framework, the primary constitutive functions are $L$2 and $L$3, with

$L$4

Input Convex Neural Networks are used so that thermodynamic admissibility is guaranteed by construction, and entropy is inferred internally even though temperature is treated as the observable field (Holthusen et al., 30 Mar 2026).

For equations of state, one approach learns a generating function $L$5 and derives

$L$6

thereby guaranteeing the first law and the integrability condition through a scalar potential. A second, more geometric approach learns a symplectomorphism of thermodynamic phase space, so that Lagrangian submanifolds are mapped to Lagrangian submanifolds and energy consistency is preserved under deformation of the manifold (Kevrekidis et al., 2024). This suggests that thermodynamic structure-informed learning can be formulated either through potentials or through geometry-preserving transformations.

5. Benchmarks and empirical behavior

The original GENERIC-based model was validated on a double thermo-elastic pendulum and on Couette flow of an Oldroyd-B viscoelastic fluid. In the pendulum, the method tracks the dynamics while conserving total energy and producing entropy as required. In the Couette-flow example, it was tested on noise-free and noisy data and compared against two ablations: an unconstrained network that omits the degeneracy loss and a black-box network that predicts the next state directly without GENERIC structure. The structure-preserving network performs best, the unconstrained network is noticeably worse, and the black-box model is worst; the method is also reported to be robust with noisy training data and able to extrapolate beyond the training time window in the viscoelastic example (Hernández et al., 2020).

In thermodynamics-aware reduced-order modeling, the Oldroyd-B Couette flow is compressed from $L$7 to $L$8 active latent variables, while a rolling hyperelastic tire is reduced from $L$9 to $M$0 active latent variables after sparsification. The paper reports very small reconstruction errors, exact qualitative preservation of $M$1 and $M$2 along the learned trajectory, and orders-of-magnitude smaller errors than an unconstrained latent predictor in several variables for the tire example (Hernandez et al., 2020).

The super-resolution extension was tested on flow over a cylinder for Newtonian and non-Newtonian fluids. The adversarial autoencoder reconstructs both low- and high-resolution fields accurately, with mean relative errors below about $M$3 for the Newtonian case and below about $M$4 for the non-Newtonian case. The autoencoder also outperforms bicubic interpolation while being about $M$5 faster in the Newtonian case and about $M$6 faster in the non-Newtonian case, and the SPNN predicts latent evolution better than a black-box neural network (Bermejo-Barbanoj et al., 2024).

A related but domain-specific example is multi-zone building thermal dynamics. There, a block-structured recurrent neural state-space model encodes heat-transfer structure, bounded temperatures, and dissipativity through inequality penalties and a Perron-Frobenius eigenvalue parameterization. On a real-world office building with $M$7 thermal zones, using only $M$8 days’ measurements for training, the method generalizes over $M$9 consecutive days and reports normalized open-loop MSE $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$0, denormalized open-loop error about $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$1 per output on test, dev error about $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$2 per output, and train error about $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$3 per output (Drgona et al., 2020).

The comparative PINN study sharpens a recurrent empirical theme: state reconstruction alone is not sufficient. In conservative benchmarks, all of NM-PINN, LM-PINN, and HM-PINN reconstruct trajectories, but NM-PINN fails to recover the conserved Hamiltonian reliably; for the mass-spring system, the Hamiltonian MSE is reported as approximately $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$4 for LM-PINN and approximately $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$5 for HM-PINN. In dissipative benchmarks, EIT-PINN achieves the smallest entropy-flux discrepancy and, for the damped pendulum, an entropy MSE of $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$6 (Li et al., 26 Mar 2026).

6. Conceptual issues, misconceptions, and research directions

A common misconception is that thermodynamic structure-informed models are simply another name for generic physics-informed models. The literature does not support that equivalence. In the foundational GENERIC formulation, no governing PDE residual is imposed; instead, thermodynamic admissibility is enforced through the reversible–irreversible split and the degeneracy constraints (Hernández et al., 2020). Conversely, in the later TSINN and THINN literature, thermodynamics informs the residual itself: one may penalize Euler–Lagrange equations, Hamilton’s equations, Onsager stationarity, entropy balance, or a large-deviation rate functional rather than an $$L\,\frac{\partial S}{\partial z}=0,\qquad M\,\frac{\partial E}{\partial z}=0.$$7 norm of a bare residual (Castro et al., 23 Sep 2025).

Another recurring point is that good state trajectories do not guarantee recovery of physically meaningful quantities. The 2026 comparative investigation explicitly shows that residual-based Newtonian PINNs can reconstruct system states while failing to recover key physical and thermodynamic quantities such as energy, Lagrangian, Hamiltonian, Rayleighian, entropy, entropy flux, or parameter values robustly. Structure-preserving formulations improve interpretability, thermodynamic consistency, and often noise robustness by restricting the hypothesis space with physically meaningful constraints (Li et al., 26 Mar 2026).

The field also contains genuine methodological trade-offs. The comparison between single-generator and double-generator formalisms reports that the single-generator model is often cheaper, more expressive, and in some experiments more accurate in raw MSE, whereas GENERIC is more interpretable and more robust to hyperparameter variation, reduced capacity, and smaller datasets. At the same time, GENERIC can be harder to train and may fall into a trivial dissipative solution when the degeneracy penalty dominates (Urdeitx et al., 2024). This suggests that thermodynamic inductive bias is beneficial, but the amount and form of structure must match the task.

Phase transitions define another important limit case. Potential-learning methods guarantee thermodynamic consistency on smooth regions, but smooth neural potentials may smear out or misplace discontinuities. In multiphase equations of state, graph correction can develop a spurious large jump in entropy near the phase transition, while additive graph correction performs better when a baseline model already preserves the phase structure. Symplectic correction is more flexible when the transition region itself must move, but it is more computationally expensive and requires a contact extension to recover the free-energy variable needed for hydrocode use (Kevrekidis et al., 2024).

A further line of work reverses the usual direction of inductive bias. Rather than hard-coding thermodynamics, a Siamese network trained on microscopic images from adiabatic molecular dynamics is shown to induce an order relation numerically consistent with the axioms of adiabatic accessibility, and its internal scalar representation behaves like entropy. This suggests that neural networks may sometimes discover thermodynamic order from microscopic trajectories when the learning problem is formulated as order detection rather than direct state prediction (Kuroyanagi et al., 2 Jun 2025).

Current comparative work points toward broader generalizations. The 2026 TSINN study identifies future interest in GENERIC, conservation-dissipation formalisms, and neural operators combined with thermodynamic structure (Li et al., 26 Mar 2026). Taken together, the literature indicates that thermodynamic structure-informed neural networks are evolving from a specific GENERIC-based architecture into a broader research program: neural models whose admissible laws, potentials, or residuals are shaped by conservation, dissipation, entropy production, and the geometry of nonequilibrium thermodynamics.