Physics-Informed Neural Surrogates

• Physics-Informed Neural Surrogates are deep learning models that embed physical laws into their architecture to improve accuracy and data efficiency.
• They employ hybrid loss functions that balance data fidelity with physics-based constraints, ensuring consistency with governing PDEs/ODEs.
• Applications include fluid dynamics, solid mechanics, and real-time control, showing significant speedups and error reductions over traditional methods.

Physics-Informed Neural Surrogates

Physics-Informed Neural Surrogates (PINS) are machine learning models—predominantly deep neural networks and neural operators—that are trained with hybrid objectives combining data-driven supervision and explicit enforcement of physical laws, typically through the residuals of governing partial differential equations (PDEs) or ordinary differential equations (ODEs). By incorporating physics-based knowledge as a soft or hard constraint in the learning process, physics-informed neural surrogates yield data-efficient, generalizable, and physically-consistent models for complex scientific and engineering systems.

1. Mathematical Foundations and Principles

Physics-informed neural surrogates are constructed by embedding the constraints of governing physical equations into the architecture, loss function, or training protocol of deep learning models. Core architectures include:

• Physics-Informed Neural Networks (PINNs): Feed-forward networks (or other topologies) whose parameters $\theta$ are learned to fit system observations while minimizing residuals of the governing PDE/ODEs at collocation points. For example, in 2D incompressible Navier–Stokes flow, PINNs approximate velocity and pressure by $u_\theta(x,y)$, $v_\theta(x,y)$, $p_\theta(x,y)$ and penalize both mismatch to data and to physics residuals (Wong et al., 2021).
• Physics-Informed Neural Operators (PINOs): Architectures such as DeepONet and Fourier Neural Operator (FNO) that learn solution operators mapping entire fields (functions) as inputs to outputs, with physics residuals incorporated into the loss (Goswami et al., 2022).

The typical PINN loss combines:

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

where $L_{\rm data}$ penalizes deviations from measurement data, and $L_{\rm phys}$ penalizes violations of the PDE, as computed via automatic differentiation at collocation points.

In operator frameworks, the loss is:

$$L(\theta) = L_{\rm data} + \lambda_{\rm PDE} L_{\rm PDE} + \lambda_{\rm BC} L_{\rm BC} + \lambda_{\rm IC} L_{\rm IC}$$

with $L_{\rm PDE}$ enforcing residual constraints on simulated or unlabeled input functions, and $L_{\rm BC/IC}$ imposing boundary/initial conditions (Goswami et al., 2022, Kim et al., 13 Feb 2026).

2. Architectures and Formulations

Physics-informed neural surrogates encompass several network forms, each suited to specific classes of physical systems:

• Multi-layer Perceptron (MLP) PINNs: Widely used for steady and unsteady PDEs, such as incompressible flow (Wong et al., 2021), nonlinear solid mechanics (Sunil et al., 2024), or 1D compressible gas dynamics (Roohi, 9 Dec 2025).
• Operator Networks (DeepONet, FNO, GNO): Designed for mapping arbitrary input functions (e.g., boundary conditions, permeability fields) to output solution fields via integral-kernel or branch–trunk (series) architectures. DeepONet combines branch and trunk networks to yield universal operator approximation (Goswami et al., 2022).
• Convolutional and Graph Architectures: Physics-informed convolutional NNs (PICNNs) for spatially local discretizations (e.g., stencil convolutions on FE meshes in FE-PINNs (Sunil et al., 2024)), and graph-based surrogates (e.g., for multi-physics or mesh-based systems (Bazzi et al., 29 May 2026, Shende et al., 19 May 2026)).
• Advanced Surrogates (PILNO, Residual-guided, Hybrid Surrogates): Architectures such as Physics-Informed Laplace Neural Operator (PILNO) (Kim et al., 13 Feb 2026) decouple transient (pole-residue) and steady-state (FNO) components, while residual-guided refinements post-train corrective networks at locations of large PDE violation (Cheung et al., 28 Dec 2025).

Parametric surrogates can be built by extending the input space to include physical parameters (e.g., Reynolds, Mach, angle of attack), thereby allowing learned models to generalize across a continuous family of operating regimes (Coulaud et al., 2024).

3. Training Protocols, Sampling Strategies, and Optimization

Training physics-informed surrogates involves careful design of sampling, loss regularization, and optimization:

• Random and Adaptive Collocation: Standard PINNs sample collocation points uniformly in the solution domain, but adaptive residual-based sampling can accelerate convergence and improve accuracy in boundary layers or shock regions (Cheung et al., 28 Dec 2025, Coulaud et al., 2024). In operator learning, virtual inputs (unlabeled, diverse synthetic functions) are used to enforce PDE residuals beyond the labeled regime, boosting out-of-distribution generalization (Kim et al., 13 Feb 2026).
• Loss Balancing: Competition among loss terms (data, PDE, boundary) can hinder training. Dynamic weighting strategies are common, such as gradually ramping the physics weight and monitoring unweighted PDE loss for generalization (Cheung et al., 28 Dec 2025). In PILNO, temporal-causality weighting up-weights early-time residuals to stabilize training (Kim et al., 13 Feb 2026).
• Optimizer Selection: Physics-informed losses are often highly nonconvex and sensitive to optimizer selection. Phase-based hybrid optimization (ADAM then L-BFGS) is standard (Coulaud et al., 2024, Rosofsky et al., 2022), while advanced optimizers such as MUON (quasi-Newton tailored to PINNs) have shown consistent improvements in convergence rate and final loss (Cheung et al., 28 Dec 2025).

4. Applications and Quantitative Performance

Physics-informed neural surrogates have demonstrated substantial gains over data-only benchmarks across diverse domains.

• Fluid and Aerodynamics: Order-of-magnitude reductions in velocity and pressure MSE for PINNs, robust to label noise and capable of interpolation/extrapolation across Reynolds numbers. For multi-parameter aerodynamics, PINNs can deliver field errors below $u_\theta(x,y)$0 and enable inference over a wide design space in milliseconds (Wong et al., 2021, Coulaud et al., 2024).
• Operators for Complex Physics: Physics-informed neural operators yield state-of-the-art performance on Darcy flow, Navier–Stokes, and brittle fracture phase-field problems, outperforming purely data-driven neural operators, and enabling rapid super-resolution or few-shot learning (Goswami et al., 2022, Rosofsky et al., 2022).
• Solid Mechanics: Physics-informed graph-based and FEM-integrated surrogates yield accurate predictions for both linear and nonlinear elasticity and outperform geometric or random coarsening in multigrid GNNs (Sunil et al., 2024, Bazzi et al., 29 May 2026).
• Nonlinear Systems Identification: SOLIS and related frameworks enable interpretable recovery of local dynamic coefficients (natural frequency, damping, gain) in state-dependent nonlinear ODEs, exceeding inverse PINN baselines in both accuracy and physical consistency (Mansur et al., 16 Apr 2026).
• Real-time and Control Applications: PINN surrogates in articulated soft robot control yield up to 500× inference speedup over first-principles models with sub-degree tracking error and excellent generalization to unseen payloads or orientations, thus enabling real-time nonlinear model predictive control (Habich et al., 4 Feb 2025).

Tables below illustrate test error reductions from physics integration in PINNs; e.g., for fluid flow (Wong et al., 2021):

Re	Data-only MSE_u	PINN MSE_u	Data-only MSE_p	PINN MSE_p
50	5 × 10⁻³	5 × 10⁻⁴	8 × 10⁻³	7 × 10⁻⁴
100	4 × 10⁻³	4 × 10⁻⁴	7 × 10⁻³	6 × 10⁻⁴
...	...	...	...	...

5. Extensions, Best Practices, and Limitations

Several extensions and developments have improved the flexibility, robustness, and applicability of physics-informed neural surrogates:

• Boundary-Only and Unbounded Domains: By recasting linear PDEs as boundary integral equations (BIEs), boundary-trained operator networks can solve for complex or unbounded domains with $u_\theta(x,y)$1 samples, a dramatic reduction in sampling complexity compared to volumetric PINNs (Fang et al., 2023).
• Parametric and Multi-physics Surrogates: Both MLP PINNs and operator networks can embed parametric dependencies as inputs, support scheduling across tracks or stages for complex thermal histories in additive manufacturing, and compose multiple operators for multi-fidelity and multi-physics coupling (Safari et al., 3 Feb 2025, Goswami et al., 2022).
• Model Compression and Decision-aware Benchmarks: Surrogates like PicoPINN use hierarchical parameter clustering and relation-matrix reparameterization to reduce parameter count by 90%+ with negligible loss in physical fidelity, enabling integration in real-time optimal control (Jiang et al., 7 Apr 2026).
• Limitations: Ill-conditioning among competing loss terms, sensitivity to optimizer, inability to exactly handle discontinuities or strong shocks, and theoretical open questions regarding approximation and generalization bounds all remain obstacles (Coulaud et al., 2024, Cheung et al., 28 Dec 2025). Decision-aware benchmarks demonstrate that low curve error does not necessarily reflect downstream engineering utility, highlighting the need for holistic evaluation metrics (Cieślak et al., 5 Jun 2026). For nonlinear or time-dependent or multi-domain PDEs, additional architectural or training innovations—such as curriculum or sequential models—may be needed (Safari et al., 3 Feb 2025).

6. Representative Use Cases and Quantitative Summary

Physics-informed neural surrogates are used for:

• Engineering design optimization (fluid and structural mechanics)
• Real-time control and estimation (soft robotics, vehicle dynamics)
• Parametric and operator learning in multiscale, multi-fidelity, or multiphysics systems
• Data assimilation and system identification for nonlinear ODE/PDE dynamics
• Accelerated simulation and surrogate modeling for UQ and Bayesian inversion

Key quantitative gains include:

• An order-of-magnitude reduction in test error (10×) on field variables for flow problems using physics loss regularization (Wong et al., 2021).
• 3×-10× faster convergence and reduction in test error when using transfer optimization with physics-informed models.
• Sub-degree tracking accuracy for PINN-based real-time control of complex mechanical systems, even in highly varied operational domains (Habich et al., 4 Feb 2025).
• In accelerated additive manufacturing simulation, speedups of over 8× relative to monolithic physics-informed operator surrogates, with multi-track error suppression (melt-pool errors reduced $u_\theta(x,y)$23–5%) (Safari et al., 3 Feb 2025).
• Decision-aware benchmarks reveal that physics-informed losses, while altering the trade-off landscape, do not always guarantee lower regret or constraint violations, underscoring the need for task-specific evaluation (Cieślak et al., 5 Jun 2026).

In summary, physics-informed neural surrogates form a versatile, theoretically grounded, and empirically validated class of models for simulating, controlling, and optimizing complex physical systems with enhanced accuracy, generalizability, and physical consistency, underpinning their widespread adoption across scientific machine learning and engineering design (Wong et al., 2021, Coulaud et al., 2024, Goswami et al., 2022, Sunil et al., 2024, Kim et al., 13 Feb 2026, Mansur et al., 16 Apr 2026, Cieślak et al., 5 Jun 2026).