Physics-Infused Neural Networks (PINN)

• Physics-Infused Neural Networks (PINNs) are deep learning models that embed differential equations and physical constraints into the loss function to ensure predictions adhere to physical laws.
• They leverage techniques like residual loss regularization, finite-difference approximations, and architecture adaptations to improve convergence and extrapolation in complex systems.
• PINNs have been effectively applied in power electronics, materials modeling, and computational finance, offering enhanced control, accuracy, and integration with traditional simulation methods.

A Physics-Infused Neural Network (PINN) is a neural network framework in which physical laws—typically in the form of differential equations, known system constraints, or analytic surrogates—are embedded within the learning process through regularization of the loss function or architecture modifications. The PINN paradigm enables neural models to synthesize measurement data with governing models, enforcing physical consistency, improving stability and extrapolation, and enabling integration with classical model-based control or simulation pipelines. In recent developments, PINNs have seen their architecture, loss design, application scope, and integration strategies systematically refined for diverse scientific, engineering, and control applications (Mahmud, 22 Mar 2026, Kajiura et al., 2023, Gupta et al., 19 Jul 2025, Martínez-Esteban et al., 29 Jul 2025, Das et al., 7 Nov 2025, Wang et al., 2023, Lim et al., 25 Feb 2026, Nikolaienko et al., 2024).

1. Foundational Principles and Mathematical Framework

PINNs are constructed by augmenting standard neural network regression—minimizing mean-squared error between predictions and observation—with additional loss terms penalizing violations of physical laws or constraints. Let $x$ be the independent variable(s), $u_\theta(x)$ the NN output, and $\mathcal{N}$ the governing operator (PDE or ODE). The canonical PINN loss is

$$\mathcal{L}(\theta) = \mathcal{L}_{\text{data}}(\theta) + \lambda_{\rm phys}\,\mathcal{L}_{\text{phys}}(\theta)$$

where

• $\mathcal{L}_{\text{data}}$ penalizes data misfit (e.g., $\|u_\theta(x_i) - y_i\|^2$ over observed data),
• $\mathcal{L}_{\text{phys}}$ penalizes violation of the governing equations (e.g., $\|\mathcal{N}[u_\theta](x_j)\|^2$ at collocation points),
• $\lambda_{\rm phys}$ is a tunable hyperparameter (Gupta et al., 19 Jul 2025, Martínez-Esteban et al., 29 Jul 2025, Lim et al., 25 Feb 2026).

Boundary and initial conditions are incorporated analogously as additional quadratic penalties (Martínez-Esteban et al., 29 Jul 2025, Gupta et al., 19 Jul 2025). For systems with unknown parameters, joint optimization over $\theta$ (NN weights) and model parameters (e.g., coefficients in $u_\theta(x)$0) enables simultaneous forward- and inverse-problem solution (Kajiura et al., 2023, Gupta et al., 19 Jul 2025).

For dynamical systems, the physical residual may originate from time-discretized ODEs, PDEs, or even algebraic constraints. PINNs can use automatic differentiation to obtain derivatives required for the residual, or finite-difference stencils to compute these terms directly, trading simplicity and computational cost for flexibility (Lim et al., 25 Feb 2026).

2. Model Architecture and Physical Knowledge Embedding

PINN architectures vary but are typically fully connected feed-forward MLPs for classical applications; more recent work incorporates RNN (Kajiura et al., 2023), gating structures (Kag et al., 2023), convolutional blocks (Zhang et al., 2022), or even attention modules tailored for sharp features (Rodriguez-Torrado et al., 2021). The physics-based regularization can be implemented through:

• Direct enforcement via residue loss: The most prevalent mechanism, residuals of the governing equations are penalized at sampled domain points (Gupta et al., 19 Jul 2025, Mahmud, 22 Mar 2026).
• Finite-difference–based residuals: Derivatives required for the residual are approximated via FDM, which simplifies implementation, offers computational gains for grid-based problems, and closely matches classical engineering workflows (Lim et al., 25 Feb 2026).
• Forward–backward integration: Physics loss can include both forward and backward time discretizations to increase data efficiency or stability (Mahmud, 22 Mar 2026).
• Physics-driven architecture adaptations: Output transforms or layers that enforce asymptotic or invariance patterns—e.g., output mapping to precisely encode positivity or correct scaling—are essential for stability in some regimes (Nikolaienko et al., 2024).

The architecture of the neural network is a critical factor in performance. Automated neural architecture search frameworks (NAS-PINN, Auto-PINN) have been developed to select depth, width, activation, and skip connections best suited to the underlying physical problem, leading to significant improvements in PINN accuracy and convergence robustness (Wang et al., 2023, Wang et al., 2022). Evolutionary search and memetic optimization further generalize the hyperparameter search process for high-dimensional or meta-learning settings (Wong et al., 11 Jan 2025).

3. Loss Function Design, Training Strategies, and Optimization

The total PINN loss balances data fitting, physical regularization (ODE/PDE/BVP/IVP residuals), and, where relevant, boundary and initial state satisfaction. Additional terms can be incorporated for constraints not directly expressible as differential equations, including periodicity (using over-sampling (Leung et al., 2021)) and dynamic interior boundary anchors (Dynamical Boundary Constraints, DBC (Martínez-Esteban et al., 29 Jul 2025)).

Key factors for PINN optimization include:

• Balancing the loss terms: Careful tuning of regularization weights is necessary; imbalanced losses can result in trivial solutions or overfitting to the data or the physics, depending on their relative scales (Gupta et al., 19 Jul 2025, Martínez-Esteban et al., 29 Jul 2025, Mahmud, 22 Mar 2026).
• Integration loss vs. multi-loss: For systems with unobservable states or many ODEs, integrating through the physics model (e.g., via unrolled Runge–Kutta integration) and penalizing only the output prediction mismatch leads to a much less complex and more robust optimization landscape compared to assembling one loss per equation (Kajiura et al., 2023).
• Optimizer selection: First-order optimizers such as Adam are typical, but second-order constrained approaches (trust-region SQP) offer strong improvements for ill-conditioned problems and guarantee constraint feasibility (Cheng et al., 2024).
• Pretraining and feasibility initialization: For hard-constrained PINNs, initializing on the constraint manifold accelerates convergence and improves robustness (Cheng et al., 2024).
• Hybrid algorithms: Evolutionary and memetic approaches—combining evolutionary strategies with gradient descent—enable the optimization of both model weights and architecture or even adaptive regularization weights for the loss terms (Wong et al., 11 Jan 2025).
• Dynamic constraint enforcement: Warm-started constraint anchoring (DBC) can systematically stabilize training, especially for multiscale or oscillatory problems (Martínez-Esteban et al., 29 Jul 2025).

4. Application Domains and Integration with Classical Methods

PINNs have been deployed in diverse domains, including power electronics (Mahmud, 22 Mar 2026), materials modeling via elasticity or homogenization (Kag et al., 2023, Leung et al., 2021), option pricing in computational finance (Dhiman et al., 2023), quantum PDE solving (Markidis, 2022), heavy-ion collision modeling (Das et al., 7 Nov 2025), reaction–diffusion–transport in catalytic chemistry (Nikolaienko et al., 2024), and dynamic control (Fotiadis et al., 28 May 2025, Kim et al., 3 Aug 2025).

In power electronics, PINNs provide fast, physically consistent state predictions that inform a finite control set MPC (FCS-MPC) loop, enabling fast and robust switching control in DC–DC converters, significantly improving transient recovery and voltage ripple over standard MPC (Mahmud, 22 Mar 2026). In chemical engineering, PINNs serve as surrogates for expensive microkinetic algebraic solvers inside hybrid finite-difference schemes, provided the PINN output transformation faithfully encodes known physical asymptotes (Nikolaienko et al., 2024).

For multiscale PDE problems, classical PINNs fail without homogenization. Neural homogenization, as in NH-PINN, splits the problem into cell problems (solved by periodic PINNs), coefficient assembly (automatic differentiation and quadrature), and a coarse-scale PDE solve—yielding dramatic gains in accuracy compared to direct PINN approaches on highly oscillatory problems (Leung et al., 2021).

In computational finance, PINNs with layered, gated MLP architectures and careful boundary–physics loss balancing can accurately learn free-boundary (American option) and standard (European option) pricing surfaces, outperforming analytic and finite-difference benchmarks on real market datasets (Dhiman et al., 2023).

5. Advanced Techniques: Architecture Search, Meta-learning, and Hybrid Approaches

Optimal PINN architecture depends on the underlying operator complexity, equation type (elliptic, parabolic, hyperbolic), and solution regularity:

• Neural architecture search (NAS): Automated search reveals that shallow and wide networks with residual (skip) connections outperform deep or constant-width networks for elliptic and advection-dominated PDEs, while moderate depth is required for nonlinear transport problems (Wang et al., 2023, Wang et al., 2022, Wong et al., 11 Jan 2025).
• Evolutionary/memetic PINNs: Hybrid algorithms that combine population-based search (for both weights and architecture) with gradient descent enable robust discovery of generalizable, high-performing models, resist local minima, and discover optimal weighting for multi-term losses (Wong et al., 11 Jan 2025).
• Meta-learning: Evolutionary approaches can meta-learn initializations that quickly adapt to new equation families with minimal training (Baldwinian PINN), outperforming model-agnostic meta-learning (MAML) in out-of-distribution generalization (Wong et al., 11 Jan 2025).
• Physics-driven output transformations: Architectures can be adapted with output layers that guarantee correct scaling and asymptotic behavior, as in PINN surrogates for chemical microkinetics (Nikolaienko et al., 2024).

These advanced methods have enabled PINNs to exceed the limitations imposed by manually designed, monolithic feed-forward architectures, to systematically adapt to complex, multiscale, or parameterized problems, and to extend into new computational paradigms (e.g., quantum PINNs) (Markidis, 2022).

6. Limitations, Challenges, and Generalization

PINNs exhibit known failure modes, including slow convergence, sensitivity to the weights of physics-informed loss terms, and ill-conditioning or spurious minima in rugged optimization landscapes—especially for stiff, high-frequency, or multi-scale PDEs (Wong et al., 11 Jan 2025, Cheng et al., 2024, Martínez-Esteban et al., 29 Jul 2025, Leung et al., 2021). They are sensitive to the fidelity of the embedded physics; mis-specification of model structure or missing physics (e.g., non-modeled parasitics in power-electronic ODEs) can degrade predictive accuracy and constraint feasibility (Mahmud, 22 Mar 2026, Nikolaienko et al., 2024).

Hybrid hard-constrained optimization, dynamical constraint anchoring, and robust NAS or evolutionary search methods mitigate—but do not eliminate—these challenges. For multiscale and periodic media, full solution accuracy is restored only by explicit homogenization steps (Leung et al., 2021). In applications beyond the training distribution, or in regimes where the physical domain is outside prior parameter ranges, domain knowledge–based output mapping or careful constraint enforcement is required to ensure stability (Nikolaienko et al., 2024).

The generalization of PINN and their hybrids to more complex systems, higher dimensions, and new physical settings (e.g., on quantum computing platforms, or for coupled PDE-ODE systems) is ongoing. Success depends critically on the architecture’s alignment with underlying physics, the enforceability of hard or soft constraints, and the ability to scale architecture and training to the problem complexity and computational environment.

7. Quantitative Benchmarks and Practical Guidelines

Empirical studies consistently confirm the gain from embedding physics into the neural network:

PINN Variant or Improvement	Problem/Setting	Typical L2 Error/Benefit	Reference
PINN + FCS-MPC	Boost converter ctrl	Transient recovery halved, RMSE $u_\theta(x)$1	(Mahmud, 22 Mar 2026)
Integration-loss PINN	Battery modeling	SoC error $u_\theta(x)$2, parameter error $u_\theta(x)$3	(Kajiura et al., 2023)
FDM-PINN vs AD-PINN	Burgers/Laplace	Up to 20% lower error and $u_\theta(x)$4 faster	(Lim et al., 25 Feb 2026)
NAS-PINN/Auto-PINN	PDE solution	1–2 orders better error, rapid hyperparam tuning	(Wang et al., 2023, Wang et al., 2022)
DBC-PINN	ODEs (oscillators)	Error drops $u_\theta(x)$5 to $u_\theta(x)$6 w/ $u_\theta(x)$7	(Martínez-Esteban et al., 29 Jul 2025)
NH-PINN	Multiscale PDEs	Error $u_\theta(x)$8, vanilla PINN fails ($u_\theta(x)$9 error)	(Leung et al., 2021)
Gated PINN for options	Option pricing	12–60% RMSE improvement over analytic/FD	(Dhiman et al., 2023)
trSQP-PINN	Hard PDEs	1–3 orders lower error than penalty/ALM	(Cheng et al., 2024)

Practical guidelines include: tuning the regularization weights for each application, balancing data/physics loss, selecting shallow wide networks for smooth or elliptic problems, utilizing advanced architectures (e.g., gating, attention) for sharp or oscillatory problems, pretraining or warm starting dynamic constraints, and supplementing the loss or architecture with domain knowledge where possible (Gupta et al., 19 Jul 2025, Wang et al., 2023, Wong et al., 11 Jan 2025, Nikolaienko et al., 2024, Martínez-Esteban et al., 29 Jul 2025).

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In summary, Physics-Infused Neural Networks unify data-driven learning and physical modeling, forming a robust methodological backbone for contemporary scientific machine learning and model-based control. Their success relies on integrated loss formulation, architecture search, advanced optimization, and hybrid methods tailored to the problem physics, with ongoing research focused on scaling accuracy, reliability, and generalization across increasingly complex physical domains and control tasks.