PI-DeepONets: Physics-Informed Operator Learning

• Physics-Informed DeepONets are operator learning models that integrate the DeepONet architecture with embedded physical laws to model PDEs and dynamical systems.
• They employ a hybrid loss strategy, combining data fidelity with physics-informed regularization through automatic differentiation to enforce governing equations.
• PI-DeepONets enhance data efficiency and generalization, offering significant speedups and robust performance across a range of scientific and engineering applications.

Physics-Informed DeepONets (PI-DeepONets) are operator learning models that integrate the expressive nonlinear operator approximation of Deep Operator Networks with embedded physical constraints, enabling accurate, efficient, and physically consistent modeling of solution operators for partial differential equations (PDEs) and dynamical systems. By leveraging automatic differentiation and tailored loss functions enforcing the governing equations, PI-DeepONets address both data efficiency and generalization limits of traditional operator learning approaches while mitigating the implementation complexity and sensitivity issues of physics-informed neural networks (PINNs). Their technical and algorithmic developments center on the synthesis of data supervision, physics-informed regularization, and systematic treatment of operator function space mappings.

1. Core Architecture of PI-DeepONets

The foundation of PI-DeepONets is the Deep Operator Network (DeepONet) architecture, which learns nonlinear mappings between function spaces by decomposing the operator into a branch network and a trunk network, whose merged outputs reconstruct the target function. For an operator mapping an input function $u$ to an output function $G(u)$, the generic form is

$$\mathrm{DeepONet}(u, y) = \sum_{k=1}^p b_k(u(x_1), \ldots, u(x_m)) \, t_k(y) + b_0,$$

where $b_k(\cdot)$ are branch net outputs parameterizing the input function via its discretized sensor values, $t_k(\cdot)$ are trunk net outputs at the query point $y$, and $b_0$ is an optional bias (Chappell et al., 22 Sep 2025, Williams et al., 2024, Wang et al., 2021). The universal approximation property of DeepONet ensures that such an architecture can represent solution operators for wide classes of PDEs to arbitrary accuracy, provided sufficient neural capacity and functional basis.

In the physics-informed extension, physical laws (differential equations, conservation constraints, structural balance, etc.) are incorporated during training by automatic differentiation of the composite DeepONet output, with physics loss terms penalizing deviations from the governing equations. This enables the model to learn not only from discrete supervised data, but also from continuous physical consistency criteria (Wang et al., 2021, Goswami et al., 2022, Chappell et al., 22 Sep 2025).

2. Physics-Informed Training Objectives and Loss Formulation

The central algorithmic distinction of PI-DeepONet is the construction of a hybrid loss,

$$\mathcal{L}_{\mathrm{PI\text{-}DeepONet}} = \mathcal{L}_{\mathrm{data}} + \lambda_{\mathrm{phys}} \, \mathcal{L}_{\mathrm{phys}},$$

where $\mathcal{L}_{\mathrm{data}}$ is a supervised loss (e.g., mean-squared error on known outputs) and $\mathcal{L}_{\mathrm{phys}}$ enforces the embedded physics (e.g., PDE residuals, energy conservation, or consistency conditions), with weight $G(u)$0 (Chappell et al., 22 Sep 2025, Wang et al., 2021, Goswami et al., 2022).

Physics loss construction employs the differentiability of DeepONet's trunk outputs in the evaluation coordinates and leverages modern automatic differentiation frameworks. In generic PDE cases, a pointwise residual

$G(u)$1

is computed at sampled "collocation" points, where $G(u)$2 denotes the spatial differential operator (e.g., advection, reaction, diffusion, etc.). These residuals are batch-averaged to form the physics-informed penalty (Williams et al., 2024, Wang et al., 2021).

Variants include:

• Taylor-based physics regularization for temporal consistency (Chappell et al., 22 Sep 2025).
• Energy conservation and static equilibrium constraints for elasticity (Ahmed et al., 2024).
• Physics residuals from inverse or coupled systems (Goswami et al., 2022, Jiao et al., 2024).
• Partition penalty for trunk outputs to enforce balanced, stable mode structure (Mi et al., 17 Dec 2025).

The approach extends to settings where no paired input-output data are available, relying solely on the physics loss and initial/boundary constraints ("zero-data" regime) (Wang et al., 2021, Sevcovic et al., 2023).

3. Training Protocols, Transfer Learning, and Knowledge Distillation

Standard training of PI-DeepONet uses gradient-based optimizers (Adam, SOAP, etc.) over mini-batches of sampled function pairs and collocation points, with hyperparameters including branch/trunk network depths, widths, sensor placement, and relative weights for data and physics losses (Karampinis et al., 7 Nov 2025, Chappell et al., 22 Sep 2025). Adaptive loss weighting and batch sampling are used to control scale imbalance and capture complex dynamics (Williams et al., 2024, Liu et al., 13 Jan 2026).

Recent developments emphasize:

• Stagewise pipelines in which a PI-DeepONet is pre-trained and then used for knowledge distillation, acting as a frozen operator-level supervisor for lightweight student networks, dramatically reducing architectural complexity and implementation burden while maintaining or improving performance (Chappell et al., 22 Sep 2025).
• Distributed pretraining and zero-shot fine-tuning: branch and trunk sub-nets are pretrained on disjoint operator families or different solution manifolds, with downstream adaptation via physics-informed loss ("PI-LoRA," "PI-Full") (Zhang et al., 2024).
• Transfer learning across parameter regimes or even between different PDE types, shown to halve final errors and restore rich functional basis spectra in challenging, "hard" cases (Williams et al., 2024).
• Hard-constraint formulations, embedding initial and boundary data exactly via mask functions, eliminate the need to learn these conditions and guarantee continuity under repeated time stepping (Brecht et al., 2023).

4. Theoretical Guarantees, Basis Interpretability, and Spectral Structure

Rigorous error bounds and basis analysis have been developed for PI-DeepONets. Under regularity and stability assumptions, PI-DeepONets achieve convergence rates with only polynomial dependence on ambient or intrinsic dimension—the curse of dimensionality (CoD) is mitigated (Ryck et al., 2022). Specifically, for solution operators of nonlinear parabolic or elliptic PDEs,

$G(u)$3

for some exponent $G(u)$4, with branch/trunk network width $G(u)$5 and depth $G(u)$6, where $G(u)$7 is the number of modes.

Analysis of the learned trunk features via singular value decomposition shows that physics-informed training yields low-dimensional, nearly universal bases—even across operator classes—leading to efficient spectral or model reduction schemes and enabling ODE integration in the learned subspace. Rapid trunk singular-value decay and expansion coefficient decay serve as sensitive diagnostics of training quality (Williams et al., 2024). Transfer learning across parameter or equation families preserves and enhances this spectral efficiency.

5. Applications in Scientific and Engineering Domains

PI-DeepONets have been deployed in diverse domains:

• Personalized Hemodynamic Modeling: PI-DeepONet learns mappings from wearable waveform segments to blood pressure signals, enforcing physiologically consistent evolution via a Taylor-based physics penalty and enabling knowledge distillation pipelines with reduced hyperparameter complexity and resource cost (Chappell et al., 22 Sep 2025).
• Hydraulics and Environmental Modeling: PI-DeepONet surrogates for 2D shallow water equations deliver order-of-magnitude acceleration versus numerical solvers, with improved out-of-distribution robustness but slight accuracy tradeoff in-distribution. Adaptive multi-component loss balancing and detailed normalization strategies are required (Liu et al., 13 Jan 2026).
• Structural Mechanics: PI-DeepONets leveraging equilibrium or energy balance via stiffness matrices achieve rapid, real-time prediction of responses for high-dimensional systems, with optional master/slave reductions via Schur complement and tailored output network strategies for multi-DOF cases (Ahmed et al., 2024).
• Traffic State Estimation: By embedding macroscopic conservation laws, PI-DeepONets achieve superior estimation from sparse observation data and improved physical plausibility compared to pointwise and purely data-driven baselines (Li et al., 18 Aug 2025).
• Porous Media, Inverse and Graph Problems: Applications include coupled FEM+PI-DeepONet frameworks for time-dependent convection-diffusion, operator learning on graphs with edge-based surrogates and domain decomposition, point-cloud manifolds, and Bayesian inversion with large-scale MCMC, exploiting PI-DeepONet’s rapid forward inference (Kara et al., 27 Aug 2025, Blechschmidt et al., 7 May 2025, Jiao et al., 2024).

PI-DeepONets also generalize efficiently in long-time integration of evolution equations, parameter variations for both initial and boundary data, and nonlinear systems, including challenging highly nonlinear dynamical regimes (Wang et al., 2021, Luan et al., 25 May 2025).

6. Limitations, Implementational Challenges, and Recent Augmentations

Key practical and theoretical challenges include:

• Hyperparameter sensitivity: Selection and balancing of physics/data loss weights, collocation densities, and architectural size remains manual or heuristic; adaptive or automated balancing is an open direction (Liu et al., 13 Jan 2026, Williams et al., 2024).
• Error accumulation and stability: For long-time integration and high-dimensional or stiff systems, error growth can occur; domain decomposition and adaptive time stepping partially address this (Wang et al., 2021).
• Expressivity in extreme regimes: Extreme parameter regimes (e.g., high nonlinearity, multiscale) may require trunk expansion, hybridization with observation data, or PoU-inspired regularization to avoid mode collapse, spectrum truncation, or large loss landscape curvature (Mi et al., 17 Dec 2025, Luan et al., 25 May 2025).
• Computational cost: While inference is typically orders of magnitude faster than traditional numerical simulation, training remains computationally intensive, due in part to automatic differentiation overhead for physics residuals and the potential need for deep or overparameterized architectures (Goswami et al., 2022, Liu et al., 13 Jan 2026).
• Boundary/initial condition enforcement: Hard-constraint ansatz provide exact satisfaction and continuity, but may require task-specific mask engineering (Brecht et al., 2023).

Extensions under active investigation include additional regularizations (partition penalty, local support functions), multi-operator transfer via distributed pretraining and zero-shot adaptation, and domain decomposition for spatial or graph-based coupling (Zhang et al., 2024, Mi et al., 17 Dec 2025, Blechschmidt et al., 7 May 2025).

7. Quantitative Benchmarks and Performance

Benchmark results reported across application domains consistently indicate that PI-DeepONet outperforms vanilla DeepONet and, in many cases, PINNs, particularly when labeled data are scarce or extrapolation to out-of-distribution inputs is required. Typical error reductions are 1–2 orders of magnitude versus purely data-driven DeepONet for the same architecture (Wang et al., 2021, Williams et al., 2024). Training with as little as 10% of observations (or even zero data) is often sufficient to match or surpass full-data models when physics is enforced (Jiao et al., 2024, Wang et al., 2021).

Operator-based supervision models distilled from PI-DeepONet supervisors reach parity with adversarial and contrastive physics-informed temporal networks at a fraction of the complexity, reducing active hyperparameters from eight or more to just two (Chappell et al., 22 Sep 2025). Speedups at inference reach 10–10,000× over classical simulation, often with sub-centisecond prediction times for high-dimensional or time-resolved fields (Liu et al., 13 Jan 2026, Ahmed et al., 2024, Karampinis et al., 7 Nov 2025).

Ablation studies confirm the necessity of physics penalties for robust training and accuracy: disabling physics-informed loss terms causes marked degradation in both test correlation and RMSE, with errors increasing by factors of 2–5 or more (Chappell et al., 22 Sep 2025, Jiao et al., 2024, Williams et al., 2024).

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• (Chappell et al., 22 Sep 2025) Physics-Informed Operator Learning for Hemodynamic Modeling
• (Williams et al., 2024) What do physics-informed DeepONets learn? Understanding and improving training for scientific computing applications
• (Mi et al., 17 Dec 2025) PIP$G(u)$8 Net: Physics-informed Partition Penalty Deep Operator Network
• (Wang et al., 2021) Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets
• (Goswami et al., 2022) Physics-Informed Deep Neural Operator Networks
• (Kara et al., 27 Aug 2025) Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources
• (Karampinis et al., 7 Nov 2025) Neural Operators for Power Systems: A Physics-Informed Framework for Modeling Power System Components
• (Liu et al., 13 Jan 2026) Physics-Informed Deep Operator Learning for Computational Hydraulics Modeling
• (Zhang et al., 2024) DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning
• (Brecht et al., 2023) Improving physics-informed DeepONets with hard constraints
• (Mi et al., 17 Dec 2025) PIP$G(u)$9 Net: Physics-informed Partition Penalty Deep Operator Network
• (Blechschmidt et al., 7 May 2025) Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification
• (Ahmed et al., 2024) Physics-informed DeepONet with stiffness-based loss functions for structural response prediction
• (Li et al., 18 Aug 2025) Physics-informed deep operator network for traffic state estimation
• (Luan et al., 25 May 2025) Physics-Informed Deep Learning for Nonlinear Friction Model of Bow-string Interaction
• (Wang et al., 2021) Long-time integration of parametric evolution equations with physics-informed DeepONets
• (Ryck et al., 2022) Generic bounds on the approximation error for physics-informed (and) operator learning
• (Jiao et al., 2024) Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators