Physics-Embedded Neural Computation

• Physics-embedded neural computation is the integration of physical laws and symmetries into neural architectures to enforce hard constraints and improve interpretability.
• It embeds conservation laws and operator constraints directly into network layers, ensuring efficient learning and robust generalization in complex dynamics.
• Applications span surrogate modeling for PDEs, quantum system solvers, and hybrid frameworks that combine physics-based and data-driven modeling.

Physics-embedded neural computation refers to a spectrum of modeling paradigms in which physical laws, symmetries, or variational principles are incorporated directly into the architecture, training, or functional form of neural networks. These paradigms transcend “physics-informed” soft regularization by embedding physical constraints—such as conservation laws, PDE structure, symplecticity, or constitutive relations—into the computational graph, architecture, or substrate of the neural computation itself. This approach aims to enhance physical consistency, interpretability, sample efficiency, identifiability, and cross-domain generalization while maintaining the expressivity of data-driven learning.

1. Motivation and Conceptual Foundations

Physics-embedded neural computation arises from the intersection of machine learning, computational physics, and neuromorphic engineering. Standard neural architectures—while universal function approximators—are generically physics-agnostic: their hypothesis space does not encode invariance to transformations, conservation laws, or operator structure present in physical systems (Mattheakis et al., 2019). This lack of embedded physics can yield accurate data fits but ultimately leads to extrapolation failures, poor sample efficiency, and opacity in mechanistic interpretation. Embedding physics directly restricts the hypothesis space to physically admissible models, guarantees hard constraint satisfaction (e.g., exact conservation), and allows for interpretable decomposition in terms of underlying physical laws or symmetries (Mattheakis et al., 2019, Barber et al., 2021, Wang et al., 29 Oct 2025).

Key operational concepts include:

• Architectural Embedding: Physical laws are reflected in architecture through hard-wiring (e.g., symplectic integration, parity layers, equivariant convolutions) or custom computational graphs (Mattheakis et al., 2019, Barber et al., 2021, Xu et al., 2024, Horie et al., 2022).
• Loss-Based Constraints: Residuals, boundary specifications, or variational principles imposed via physics-based loss functions, but ultimately these serve as regularization and do not guarantee hard physical constraint satisfaction (Scellier, 2021).
• Hybridization with Physical Models: Known dynamics are separated and implemented as differentiable blocks, with neural networks learning only unmodeled or residual terms (hybrid, gray-box, or white-box modeling) (Zheng et al., 4 Aug 2025, Kim et al., 2022, Heng et al., 17 Jun 2025, Xie et al., 24 Oct 2025).
• Physical Substrate and Variational Computing: Computation and (sometimes) learning are performed by the physical system itself (e.g., memristor arrays, photonic chips, mechanical circuits), embedding laws into the substrate and its adaptation dynamics (Yu et al., 2024, Scellier, 2021).

2. Symmetry, Conservation, and Operator Embedding in Architecture

Physics embedding can be realized through explicit architectural modules or layers designed to enforce discrete or continuous symmetries, conservation laws, or operator-theoretic features.

• Symmetry Embedding: Parity (even/odd function) is enforced by constructing outputs as sum and difference of activations at mirrored inputs, employing “hub layers” to hard-code transformations (Mattheakis et al., 2019, Barber et al., 2021). Group equivariant neural networks generalize this to translation, rotation, and permutation symmetry, critical for PDEs and many-body systems (Xu et al., 2024, Horie et al., 2022).
• Conservation via Symplectic/Hamiltonian Structure: By encoding the symplectic form and Hamilton’s equations into the architecture (e.g., Hamiltonian neural networks, symplectic integrators), one guarantees energy (and momentum/volume) conservation over arbitrary function approximation (Mattheakis et al., 2019, Barber et al., 2021). Such networks drastically outperform unconstrained networks in chaotic and long-time integration regimes.
• Operator-Level Embedding: For PDEs, the network may generate outputs such that, by construction, a target operator (e.g., the convective/transport or momentum flux operator) is satisfied. Transport-Embedded Neural Networks (TENN) realize this for advection-diffusion dynamics by tensor constructions that guarantee the divergence form of the equations inside the computational graph—no residual needed (Jafari, 2024). Fourier Neural Operators with built-in momentum-preserving spectral filters enforce translation and rotation invariance, guaranteeing strict conservation laws via Noether’s theorem (Xu et al., 2024).
• Boundary Condition Enforcement: In mesh-based or graph neural networks, boundary conditions may be imposed exactly at each iteration by boundary-layer projections or equivariant message passing, with specialized layers for Dirichlet or Neumann constraints (Horie et al., 2022).

Symmetry/Constraint	Example Method	Guarantee Type
Parity (even/odd)	Hub/parallel MLP layers	Hard
Energy conservation	Symplectic/hamiltonian networks	Hard
Momentum/rotation	Equivariant CNN, MC-Fourier layers	Hard
PDE operator (advection)	Transport embedding (TENN)	Hard
Boundary conditions	Boundary encoders/layers (GNN)	Hard (for Dirichlet)

3. Physics-Embedded Hybrid and Residual Modeling Frameworks

Many physical systems admit partial but not complete mechanistic descriptions. Physics-embedded computation leverages these “gray-box” regimes by:

• Hybrid ODE/PDE Solvers: Known physical terms are implemented as exact differentiable function blocks; a compact neural network supplements as a residual, learning only the incomplete or unmodeled component. PENODE for hybrid power systems structurally embeds both the physical vector field (linear ODE blocks parameterized as state-space matrices) and a neural MLP for the residual drift, capturing sim-to-real transitions and providing interpretability (Zheng et al., 4 Aug 2025). Similarly, PENN for biomechanical motion estimation embeds a differentiable Hill-type musculoskeletal model as a fixed physics block, with a CNN trained only on the modeling mismatch (Heng et al., 17 Jun 2025). The event automaton module in PENODE accurately captures discrete mode switches critical for hybrid dynamics (Zheng et al., 4 Aug 2025).
• Physics-Integrated PDE Solvers: Neural networks may be embedded within the PDE update equations themselves, forming a closure or parameterization model whose gradients are backpropagated through the full solver via AD. This allows network weights to be optimized even without direct labels for unknown quantities (e.g., velocity-dependent viscosity in Navier–Stokes), enabling closure learning with only field observations (Iskhakov et al., 2020).
• Neural Constitutive Laws in Continuum Mechanics: Simulators for elastoplastic materials employ dual-network architectures (for elasticity and plasticity), with parameter vectors (e.g., modulus, yield strength) conditionally input to the MLPs; physics constraints (e.g., frame indifference, plastic yield) are enforced by the embedding of the MLPs within the discrete time-stepping routine, such as in Material Point Methods in PC-NCLaws (Xie et al., 24 Oct 2025).

4. Variational Principles and Physics-Based Learning Dynamics

A deep theoretical foundation for physics-embedded computation is the direct expression of neural computation as a variational (energy/action) minimization process (Scellier, 2021).

• Equilibrium Propagation: For energy-based networks, physical equilibrium $s^\star$ is chosen to minimize an energy functional $E(\theta, x, s)$, with physical or data-encoded costs incorporated as small perturbations. The fundamental learning rule extracts the true loss gradient by measuring the parameter-dependent change in the physical energy when lightly nudged toward the target—compatible with in-situ gradient measurement on physical substrates (e.g., electrical circuits, optical systems, mechanical lattices) (Scellier, 2021, Yu et al., 2024).
• Physical Self-Learning: In bottom-up physical neural networks (PNNs), adaptation of material properties (memristor conductance, magnetic domain walls, photonic interference coefficients) encodes learning governed by local, physics-prescribed rules—Hebbian adaptation, Oja forgetfulness, spike-timing dependent plasticity, or physical instantiations of contrastive learning and equilibrium propagation (Yu et al., 2024). These realizations offer non-von-Neumann, energy-efficient, and parallel updates that are “physics-native”.
• Lattice Field-Theoretic Neural Computation: A formal framework connects neural population activity, interactions, and memory via lattice field theory, mapping neural variables to field actions with spatial and temporal coupling matrices. Renormalization group methods yield scale-bridging, generative models and provide explicit connections to Boltzmann machines, Hopfield nets, recurrent dynamics, and temporal attention (Bardella et al., 2024).

5. Applications and Generalization

Physics-embedded neural computation enables robust, generalizable simulation and inverse modeling across high-dimensional and multiphysics domains:

• Spatiotemporal PDE Modeling and Inverse Law Discovery: Hierarchical Physics-Embedded Learning frameworks, such as HPE-AFNO, structurally decouple (i) fundamental operators (e.g., gradient, Laplacian, mobility) via AFNO blocks and (ii) governing combinations, allowing explicit insertion of known physics and interpretable symbolic regression for model discovery. White-box, black-box, and gray-box regimes are seamlessly unified, achieving state-of-the-art predictive and extrapolative accuracy for challenging nonlinear PDEs (Cahn–Hilliard, Allen–Cahn, Ginzburg–Landau, KPZ) (Wang et al., 29 Oct 2025).
• Surrogate Modeling in Inverse Problems: In scanning electron microscopy, a physics-embedded framework couples a differentiable electron-optical forward model and Vision Field Transformer surrogate with a self-supervised, closed-loop optimization; this enforces geometric and photometric consistency, yields metrologically accurate nanoscale reconstructions, and eradicates global drift—demonstrating adaptivity to other ill-posed inverse problems (Wang et al., 12 Jan 2026).
• Quantum System Solvers: In quantum mechanics, embedding Schrödinger equation residuals or process-driven energy recurrences within the neural architecture enables joint eigenstate/eigenvalue learning, surpassing vanilla PINNs in accuracy and stability, and directly enforcing orthogonality, normalization, and spectral structure (Wu et al., 30 May 2025).
• Uncertainty Quantification and Physics-aware Bayesian Inference: Inputs encoding nuclear shell structure in energy-dependent fission yield prediction, coupled with rigorous hyperparameter selection (WAIC), yield physically consistent, uncertainty-quantified posterior distributions (Chen et al., 24 Apr 2025).

6. Limitations, Open Challenges, and Future Directions

Despite broad success, current limitations and research challenges persist:

• Scalability: While embedding discretized operators and symmetries is tractable for moderate networks and mesh sizes, scaling to high-dimensional PDEs, large graphs, or complex physics (e.g., fully turbulence-resolving DNS, strongly coupled multiphysics) increases computational cost—a factor of $\sim 2\times$ or more per epoch is observed for operator-embedded architectures (Jafari, 2024).
• Compositionality of Constraints: Combining multiple physical constraints (e.g., simultaneous energy and momentum conservation, multiple symmetries) in a single architecture is non-trivial and presents open challenges in automating architecture generation for arbitrary constraint sets (Mattheakis et al., 2019).
• Generalization to Arbitrary Domains and Conditions: Transfer across new geometries, boundary conditions, or parameter regimes is promising (e.g., PC-NCLaws, physics-embedded GNNs), but cross-domain robustness under unmodeled physics or insufficient physics embedding remains an active area.
• Physical Self-Learning and Unsupervised Adaptivity: Natively physical learning systems achieve remarkable efficiency and autonomy but are limited in scale, architectural complexity, and measurement access. Extending mechanisms such as equilibrium backpropagation or contrastive local learning to large, deep, or temporally extended physical systems remains a significant technical barrier (Yu et al., 2024).
• Integration with Real-time and Edge Devices: Physics-embedding aids quantization and miniaturization for FPGA, neuromorphic, and embedded deployments, with event-driven or hybridized models demonstrating large reductions in neuron count and energy—but achieving high fidelity with minimal parameter budgets and across complex switching or hybrid regimes is under ongoing development (Zheng et al., 4 Aug 2025, Garg et al., 8 Nov 2025).
• Interpretability and Meta-Learning: Explicit interpretability is a strength of symbolic and operator-level embedding, but automating the extraction of meaningful analytical laws remains an open challenge in symbolic regression and meta-learning (Wang et al., 29 Oct 2025).

7. Summary Table of Key Approaches and Exemplars

Embedding Mechanism	Prototypical Methods/Examples	References
Symmetry constraints	Parity hub layers, equivariant GNN/CNN	(Mattheakis et al., 2019, Horie et al., 2022, Xu et al., 2024)
Conservation laws	Symplectic/HNN, MC-Fourier, TENN	(Mattheakis et al., 2019, Barber et al., 2021, Xu et al., 2024, Jafari, 2024)
Operator embedding	TENN, PeFNN, AFNO	(Jafari, 2024, Xu et al., 2024, Wang et al., 29 Oct 2025)
Boundary encoding	Boundary-layer projections in GNN, NIsoGCN	(Horie et al., 2022)
Hybrid residual models	PENODE, PENN, PC-NCLaws	(Zheng et al., 4 Aug 2025, Heng et al., 17 Jun 2025, Xie et al., 24 Oct 2025)
Variational/physical	EqProp, memristive/physical substrate NNs	(Scellier, 2021, Yu et al., 2024)
Physics-Integrated solvers	PDE-embedded NNs, inverse closure	(Iskhakov et al., 2020)
Physics-driven UQ	PE-BNN with shell embedding, WAIC tuning	(Chen et al., 24 Apr 2025)

Physics-embedded neural computation establishes a multi-layered formalism for integrating deep learning models with the laws of nature, offering hard guarantees on physical consistency, enhanced generalization, interpretable structure discovery, and adaptive deployment from HPC to edge and neuromorphic hardware. Continuing progress will depend on advances in architecture composition, scalable physical optimization, and the development of new modalities of physical learning and in situ adaptation.