Hybrid Physics-Informed Models

• Hybrid Physics-Informed Models are computational frameworks that combine explicit physical laws with data-driven methods to solve forward and inverse problems.
• They use strategies such as additive decomposition, convex blending, and embedded physics to integrate numerical solvers (e.g., FEM) with neural networks.
• These models enhance accuracy, generalization, and interpretability in applications ranging from high-frequency PDEs to control systems while managing computational costs.

Hybrid physics-informed models refer to computational frameworks that integrate explicit physical knowledge with data-driven (machine learning or deep learning) components, typically to improve the accuracy, generalization, stability, and interpretability of solutions to forward and inverse problems in science and engineering. The hybrid paradigm leverages complementary strengths: physics-based components enforce known constraints and guarantee plausible extrapolation, while data-driven modules flexibly model unknown, nonlinear, or high-dimensional dependencies where first-principles descriptions are inaccessible or infeasible. The hybrid methodology has been instantiated in a multitude of architectural, algorithmic, and application-specific forms across dynamical systems, partial differential equations (PDEs), control, and prediction tasks.

1. Principles and Taxonomy of Hybrid Physics-Informed Models

Hybrid physics-informed models are built on the principle of compositionality: they explicitly partition the modeling domain or task into subproblems where physical laws are enforced and subproblems where data-driven inference provides maximal benefit. Several canonical strategies can be identified:

• Additive/Corrective Decomposition: The state equation is split into a physically derived part and a data-driven correction term, e.g., $$N[u] = f_{\text{phys}}(u;\theta_\text{phys}) + f_{\text{ML}}(u;\theta_\text{ML})$$ (Blakseth et al., 2022).
• Convex (Trainable) Blend: Parallel architectures compute independent physical and neural estimates and produce a convex combination with a learned mixing weight (Huang et al., 14 Nov 2025).
• Embedded Physics Incorporation: Physical surrogates (e.g., reduced-order models, empirical laws) provide features to internal layers of a neural architecture, guiding representations (Pawar et al., 2021).
• Domain/Task Decomposition: The problem domain is spatially or temporally partitioned; classical solvers (e.g., finite element/volume methods) are used on subdomains (e.g., near boundaries), neural networks in others, with coupling at interfaces (Sobh et al., 14 Jan 2025, Xiang et al., 2022).
• Cooperative or Adversarial Mutual Regularization: Physics-based solvers and neural networks are trained together, with the interaction term penalizing discrepancies in predictions over the domain (Liverani et al., 27 Feb 2026, Liverani et al., 17 Sep 2025).

This taxonomy is non-exhaustive: the hybrid concept subsumes operator-theoretic, variational, and game-theoretic couplings; single-level and bilevel optimization; monolithic or modular implementations.

2. Model Architectures and Coupling Mechanisms

A broad diversity of hybrid architectures has been developed in recent literature. Key examples include:

• Hybrid Parallel Kolmogorov–Arnold–MLP Networks: The modified HPKM-PINN architecture (Huang et al., 14 Nov 2025) comprises parallel MLP and Kolmogorov–Arnold Network (KAN) branches. The MLP captures global smooth features, while the Fourier-based KAN specializes in high-frequency details. The hybrid output is

$$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$

where $S(\alpha)$ is a squashing function mapping a trainable parameter to $[0, 1]$. Overlapping domain decomposition further partitions $\Omega$; independent local HPKMs are trained with windowing and assembled into the global solution.

• Hybrid Residual (Neural–RBF) Networks: In HyResPINNs (Cooley et al., 2024), each residual block adaptively weights contributions from a standard neural sub-network and an RBF sub-network using a trainable parameter $\alpha^{(l)}$ per block. The network self-tunes local/global expressivity, providing both meshfree adaptivity and the spectral bias mitigation needed for challenging PDEs.
• Domain-Decomposed PINN–FEM and PINN–FDM Blends: The PINN-FEM model (Sobh et al., 14 Jan 2025) imposes Dirichlet BCs by solving the PDE with a thin FE strip near the boundary and represents the solution in the interior with a DNN (PINN), coupling at the interface node. Similarly, HFD-PINN (Xiang et al., 2022) replaces automatic differentiation by finite-difference approximations in the easy/informative interior points, while defaulting to AD near boundaries.
• Corrective Source Term Hybridization: The CoSTA framework (Blakseth et al., 2022) augments a first-principles PDE with a learned (neural) corrective source term,

$$\frac{\partial u}{\partial t} - \kappa \nabla^2 u = S_\theta(x, y, t, u, \nabla u)$$

trained to bridge the bias resulting from incomplete physical closure.

• Graph–Mesh and Operator Hybridizations: PiGMeN (Chenaud et al., 2024) constructs a graph neural network architecture—inductive on the mesh connectivity—where physics-informed terms (gradients, weak forms) are computed by differentiable finite element numerical kernels, enabling consistent backpropagation in strongly irregular or parametric geometries.
• Dual-Level Sequential Couplings: Dual-level forecasting (Nasiri et al., 12 Jan 2026) first performs hybrid LSTM–state-transition modeling for input variable prediction, then feeds sampled input trajectories into PINN output predictors, decoupling stochastic input propagation from physical output prediction.
• Quantum-Classical Hybrids: QPINN-MAC (Lantigua et al., 10 Nov 2025) composes a classical MLP and a quantum circuit per output channel, with both additive and multiplicative couplings. Universal approximation for ODE solutions and gradient control (barren plateau avoidance) are formally established.

The table below summarizes selected architectures:

Model/Reference	Components and Coupling Strategy	Key Operational Domain
HPKM-PINN (Huang et al., 14 Nov 2025)	Parallel MLP + KAN, trainable mixing, overlapping domain decomposition	High-frequency PDEs
HyResPINN (Cooley et al., 2024)	Residual blocks: weighted DNN + RBF	Stiff/singular PDEs
PINN-FEM (Sobh et al., 14 Jan 2025)	Domain: FEM near boundary, NN in interior	Strong BC enforcement
CoSTA (Blakseth et al., 2022)	PBM + NN source correction	Partially-known source PDEs
PiGMeN (Chenaud et al., 2024)	GraphNet + differentiable FE kernels	Complex geometry, mesh-based

3. Optimization and Training Procedures

Training hybrid physics-informed models generally involves minimization of composite loss functions reflecting both data fidelity and physical consistency, with additional tunable, learnable, or adaptive weights. Core elements:

• Composite Loss Functions: Hybrids typically define

$$L(\theta) = \lambda_\text{data} L_\text{data} + \lambda_\text{phys} L_\text{phys} + \lambda_\text{int} L_\text{int}$$

balancing observation error, PDE (or dynamical) residuals, and cross-component interaction (e.g., agreement between physical and data-driven modules) (Huang et al., 14 Nov 2025, Liverani et al., 17 Sep 2025, Liverani et al., 27 Feb 2026).

• Adaptive and Curriculum Weighting: Learnable parameters or gate functions modulate the influence of physics and data, either through trainable $\alpha$s (Huang et al., 14 Nov 2025, Cooley et al., 2024), dynamic cosine-similarity gates to resolve gradient conflict (Golooba et al., 25 Mar 2026), or by alternation schemes (e.g., alternating minimization for Nash equilibria in multi-agent setups (Liverani et al., 17 Sep 2025)).
• Domain or Subdomain Parallelization: Overlapping domain decomposition (Huang et al., 14 Nov 2025), parallel graph-block updates (Chenaud et al., 2024), and agent-based mutual optimization (Liverani et al., 27 Feb 2026) all enable distributed and scalable training.
• Self-supervised Physics-Informed Fine-tuning: After initial data-driven pretraining, modules such as physics-infused fine-tuning blocks (Du et al., 16 May 2025) or extended Kalman filters (Fang et al., 2024) can refine representations, efficiently enforcing PDE constraints on generated trajectories or denoising observed signals.
• Data Generation and Hybridization with Simulation: Surrogate models, as in metabolic cybergenetics (Espinel-Ríos et al., 2024), are trained using carefully curated simulation data (e.g., flux balance analysis on metabolic networks), followed by embedding into dynamic ODE frameworks.

4. Benchmark Problems, Empirical Results, and Comparative Analyses

Hybrid physics-informed models have been systematically benchmarked against pure data-driven and pure physics-based baselines across diverse settings:

• Multiscale and High-Frequency PDEs: In the 2D Helmholtz benchmark (Huang et al., 14 Nov 2025), the modified HPKM-PINN (using overlapping domain decomposition and trainable mixing) achieved normalized $L_2$ errors $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$0 ($$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$1), outperforming pure KAN ($$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$2) and MLP ($$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$3) with comparable or lower training cost and no manual hyperparameter tuning.
• Nonlinear Reaction–Diffusion and Allen–Cahn: HPKM-PINN and HyResPINN architectures yield order-of-magnitude reductions in error relative to classical PINN-style models, especially in regimes with sharp internal layers or multiple frequency scales (Huang et al., 14 Nov 2025, Cooley et al., 2024).
• Complex Geometries and Boundary Conditions: Hybrid approaches relying on mesh-based (FEM (Sobh et al., 14 Jan 2025)) or graph-mesh (Chenaud et al., 2024) processing achieve high accuracy and reliable BC enforcement on domains with cracks, holes, or complex topologies, where pure PINNs degrade.
• Time Series Forecasting: The dual-level hybrid (STM + PINN) achieves MSE reductions of 1–2 orders of magnitude over both conventional STM and data-driven FFNN models in multistep chemical process forecasting (Nasiri et al., 12 Jan 2026).
• Mechanical and Control Systems: In cart–pole dynamics (Roehrl et al., 2020), hybrid PINODE merges Lagrangian mechanics with a neural correction for non-conservative forces, attaining 2–3$$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$4 lower MAE (cart position $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$5 m vs $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$6 m for ODE model).
• Safety and Planning (Autonomous Vehicles): Hybrid models integrating LLMs with physical reasoning (e.g., social-force trajectory generation) enforced rigorous surrogate safety metrics to reduce collision rates by $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$7 compared to baseline social force and learned trajectory models (Gan et al., 6 Apr 2025).

These results consistently demonstrate that hybridization leads to substantial improvements in accuracy, robustness under noise or data sparsity, and physical plausibility versus naively trained ML or simplified physics-only solvers.

5. Generalization, Robustness, and Limitations

Hybrid physics-informed models address central challenges faced by pure ML and pure physics-based approaches:

• Generalization and Extrapolation: By enforcing physical constraints globally or through localized corrections, hybrids inherit the extrapolation stability of physical models even in out-of-distribution regimes or with sparse/noisy data (Blakseth et al., 2022, Liverani et al., 17 Sep 2025).
• Interpretability and Trustworthiness: Explicit embedding or cooperation of physics-based modules renders network predictions physically meaningful, enabling "sanity checks," interpretability of corrections, and direct improvement of mechanistic submodels (Blakseth et al., 2022, Pawar et al., 2021).
• Computational Cost and Scalability: The principal limitation is doubled model complexity and the need for careful algorithmic design to retain efficiency (especially for alternation or parallel schemes) (Liverani et al., 17 Sep 2025, Liverani et al., 27 Feb 2026). Convergence theory for non-convex or high-dimensional hybrids remains open (Liverani et al., 27 Feb 2026).
• Hybridization Overheads and Design Choices: Practitioners must balance architectural complexity (e.g., number of branches, mixing strategies), select domain splits and windowing carefully (Huang et al., 14 Nov 2025), and be mindful of limits to surrogate correction expressivity.

6. Applications and Prospects

Hybrid approaches are adaptable across disciplines:

• Scientific Computing: Turbulence modeling, uncertainty quantification in climate modeling, inverse problems in subsurface flow (Liverani et al., 17 Sep 2025, Chenaud et al., 2024), kinetic theory (e.g., collisionless Boltzmann) (Heravifard et al., 12 Dec 2025).
• Engineering Design and Digital Twins: Aerodynamics (lift prediction) (Pawar et al., 2021), wind energy, and adaptive digital twin frameworks integrating fast PBM surrogates and ML closures (Pawar et al., 2021).
• Systems and Control: Satellite attitude control (Cena et al., 17 Feb 2026), model predictive control in bioprocesses (Espinel-Ríos et al., 2024), vehicle dynamics (Fang et al., 2024).
• Epidemiology and Biomedicine: Data-physics blending in stiff multiscale models and robust parameter estimation via gradient-gated training (Golooba et al., 25 Mar 2026), population dynamics (Lalic et al., 2024).
• Autonomous Systems and Planning: Safety-critical trajectory planning fusing LLM-based scene analysis with physics (social forces) (Gan et al., 6 Apr 2025).

Ongoing research focuses on theoretical guarantees regarding convergence and stability in nonlinear non-convex hybrids, the integration of operator-based models (e.g., neural operators), privacy-preserving and federated multi-agent hybrids, and adaptive hybridization for dynamic real-time systems.

7. Table of Key Models and Their Features

Model & Reference	Hybrid Structure	Physics Module(s)	Data Module(s)	Domain Coupling	Application Area
HPKM-PINN (Huang et al., 14 Nov 2025)	KAN + MLP, trainable mixing $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$8	Fourier-based KAN	MLP	Overlapping domain	Multiscale, high-freq PDEs
HyResPINN (Cooley et al., 2024)	Residual DNN–RBF with adaptive $$u_\text{hybrid}(x; \theta) = S(\alpha) u_\text{KAN}(x; \theta_\text{KAN}) + [1-S(\alpha)] u_\text{MLP}(x; \theta_\text{MLP})$$9	RBF block	DNN block	Layerwise adaptive	Heterogeneous PDEs
CoSTA (Blakseth et al., 2022)	PBM + DNN source correction	FVM PDE (PBM)	Feed-forward DNN	Operator Injection	Heat with unknown sources
PINN-FEM (Sobh et al., 14 Jan 2025)	FEM (boundary) + PINN (interior)	Linear/tria FE	PINN	Hard interface node	Elasticity, strong BCs
PiGMeN (Chenaud et al., 2024)	GNN with differentiable FE kernels	FE kernels	GNN	Mesh-to-graph	Complex geometry PDEs
HYCO (Liverani et al., 27 Feb 2026)	Alternating physical+synthetic agents	PDE solver	Neural net	Mutual regularization	Inverse, forward PDEs

In summary, hybrid physics-informed modeling constitutes a robust framework enabling data-driven models to exploit physical laws for constraint enforcement, extrapolation stability, and interpretability, while simultaneously harnessing the expressive capacity and adaptivity of flexible neural architectures for high-dimensional or partially-understood phenomena. This double paradigm is empirically proven to deliver substantial gains in accuracy, efficiency, and generalization across a broad spectrum of complex scientific and engineering problems.