PI-DeepONet: Physics-Informed Deep Operator Networks

Updated 7 August 2025

PI-DeepONet is a neural operator architecture that approximates complex nonlinear mappings between infinite-dimensional spaces while enforcing physical laws via PDE residuals.
Its composite loss function combines data fidelity with automatically differentiated physics constraints, resulting in superior accuracy and data efficiency.
The framework is applied to parametric PDEs, inverse problems, and real-time multiphysics simulations, significantly reducing computational costs compared to traditional methods.

Physics-Informed Deep Operator Networks (PI-DeepONet) are neural operator architectures that approximate nonlinear mappings between infinite-dimensional function spaces, while simultaneously enforcing the governing physical laws of the target system. By harnessing operator learning together with physics-based constraints, PI-DeepONet achieves superior data efficiency, accuracy, and generalization in modeling and solving parametric partial differential equations (PDEs) and related scientific computing tasks.

1. Core Principles and Architecture

PI-DeepONet builds on the Deep Operator Network (DeepONet) framework, in which the solution operator $\mathcal{G}$ mapping from input functions to solution fields is decomposed as

$\mathcal{G}_\theta(u)(y) = \sum_{k=1}^p b_k(u(x_1),\ldots,u(x_m))\, t_k(y)$

where $b_k$ are outputs of the branch network that encode the discretized input function (through sensor values at $\{x_j\}$ ), and $t_k(y)$ are outputs of the trunk network evaluated at coordinates $y$ (such as space-time points) (Wang et al., 2021). This construction provides nonlinear operator approximation between infinite-dimensional spaces.

To integrate physics, the loss function is augmented by a physics constraint term enforcing residuals of the governing PDE and associated initial/boundary conditions. The composite loss is

$\mathcal{L}(\theta) = \mathcal{L}_{\mathrm{data}}(\theta) + \mathcal{L}_{\mathrm{physics}}(\theta)$

with $\mathcal{L}_{\mathrm{physics}}$ constructed using automatic differentiation of the DeepONet output to compute derivative residuals with respect to the PDE. For example, for a time-dependent PDE $u_t + \mathcal{N}(u) = f$ , the physics-term is

$\mathcal{L}_{\mathrm{physics}} = \sum_{i,j} \left| \frac{\partial}{\partial t} \mathcal{G}_\theta(u^{(i)})(x_j) + \mathcal{N}(\mathcal{G}_\theta(u^{(i)}))(x_j) - f(x_j) \right|^2$

Quantities in the loss are evaluated at collocation points within the domain. This enforces that the predicted solution functions not only minimize the mismatch with observed/simulated data, but also satisfy the residuals of the governing PDE throughout the input domain (Wang et al., 2021, Goswami et al., 2022, Sevcovic et al., 2023).

PI-DeepONet generalizes this approach for parametric, nonlinear, and high-dimensional PDEs by imposing the physical law at the operator level, and can often work in regimes where no paired input–output data are available beyond initial or boundary conditions.

2. Methodological Advances and Training Strategies

Several methodological advances underpin state-of-the-art PI-DeepONet:

Automatic Differentiation for Physics Loss: Spatial and temporal derivatives required for enforcing PDE residuals are computed via automatic differentiation, ensuring differentiable loss construction without manual discretization (Wang et al., 2021, Kushwaha et al., 21 Mar 2024).
Composite Loss Functions: Typically, the total loss includes data term(s) (e.g., pointwise $\ell_2$ error on labeled data), physics residuals, and penalty terms for violation of initial/boundary constraints. Self-adaptive weights are sometimes used to optimize the trade-off among these terms (Goswami et al., 2022).
High-Dimensional Scalability (Separable Architectures): The curse of dimensionality is addressed by separable operator learning, where each trunk sub-network handles a 1D coordinate and the full multidimensional basis is constructed as an outer product, enabling linear rather than exponential scaling in memory and computational cost (Jiao et al., 7 Jul 2024, Mandl et al., 21 Jul 2024).
Multi-Operator and Multifidelity Extensions: Networks can be pretrained on diverse operator datasets ("distributed/federated pretraining") and fine-tuned via physics loss for rapid adaptation to new PDE tasks ("zero-shot" learning) (Zhang et al., 11 Nov 2024). Multifidelity architectures combine low- and high-fidelity data and corrections to reduce data requirements (Howard et al., 2022).
Edge-Based Domain Decomposition: For metric graph and network PDEs, edge-type-specific DeepONets are coupled via additional vertex penalty losses enforcing continuity and conservation (Kirchhoff conditions) (Blechschmidt et al., 7 May 2025).

3. Applications and Performance Benchmarks

PI-DeepONet has demonstrated strong empirical performance across a wide array of scientific and engineering problems:

Parametric PDEs: Applications include ODEs (integration operators), diffusion–reaction systems, Burgers' and Navier–Stokes equations, Hamilton-Jacobi-BeLLMan, Eikonal equations, and nonlinear parabolic and semi-linear PDEs (Wang et al., 2021, Sevcovic et al., 2023, Lin et al., 2023, Mandl et al., 21 Jul 2024).
Inverse Problems and Parameter Identification: Frameworks combining PI-DeepONet with Bayesian inference/variational approaches yield robust uncertainty quantification and parameter estimation (including for spatially varying coefficients), often using sparse or noisy measurements (Kag et al., 6 Dec 2024, Raj et al., 18 Jan 2025, Jiao et al., 7 Jul 2024).
Operator Surrogates for Multiphysics and Real-Time: Advanced architectures enable near-instantaneous prediction of full-field solutions (stress, temperature, pressure, etc.) for complex manufacturing and biomedical problems (steel casting, additive manufacturing, vascular flows in aortic aneurysm), with speedups up to $4{\times}10^4$ over conventional simulation (Kushwaha et al., 21 Mar 2024, Cruz-González et al., 19 Mar 2025).
Structure-Preserving and Data Scarcity: Modified PI-DeepONets preserve conservation laws (e.g., mass in kinetic equations and collision operators) and employ entropy-inspired data sampling to focus training on physically relevant regimes (Lee et al., 26 Feb 2024).

Performance metrics consistently show that PI-DeepONet can yield $1$–$2$ orders of magnitude lower prediction errors than conventional DeepONet when labeled data is scarce, and can generalize to out-of-distribution operator inputs (Wang et al., 2021). With physical constraints enforced, PI-DeepONet shows superior robustness under data scarcity and is significantly more data-efficient compared to purely supervised neural operators.

4. Key Theoretical and Analytical Properties

The rigorous foundation for PI-DeepONet is built on approximation theory for operators and operator regression between Banach spaces (Wang et al., 2021). Key properties include:

Universal Approximation: Both vanilla and separable architectures possess universal approximation theorems for nonlinear continuous operators; i.e., for any nonlinear (possibly PDE) operator and arbitrary accuracy, there exists a PI-DeepONet realization within the architecture class (Jiao et al., 7 Jul 2024, Mandl et al., 21 Jul 2024).
Error Bounds (Approximate Operators): When applied to unknown manifolds or with approximate differential operator discretizations (Diffusion Maps, RBF, GMLS), PI-DeepONet retains provable theoretical consistency, with convergence rates quantified (e.g., for GMLS, residual error decays as $O(N^{-(p-1)/d})$ on $N$ -point clouds) (Jiao et al., 7 Jul 2024).
Model Reduction and Basis Learning: The trunk network basis functions extracted from a well-trained PI-DeepONet can serve as custom spectral/model reduction bases for the underlying PDE operator. The decay of basis singular values and coefficients provides a quantitative measure of the expressive power learned (Williams et al., 27 Nov 2024).
Transfer Learning and Multi-Operator Adaptation: Pretraining on a family of related operators or PDE tasks substantially reduces fine-tuning error and enables zero-shot adaptation; this is particularly effective when combined with physics-informed fine-tuning losses (Williams et al., 27 Nov 2024, Zhang et al., 11 Nov 2024).

5. Impact, Limitations, and Generalization

PI-DeepONet has had substantial impact on operator regression in computational science and engineering:

Data Efficiency and Robust Generalization: By leveraging physical laws, PI-DeepONet reduces the reliance on extensive labeled data, and generalizes robustly even in highly nonlinear, high-dimensional, or data-scarce settings.
Computational Effectiveness: Once trained, inference is orders of magnitude faster than traditional numerical PDE solvers; this enables real-time computation in digital twins, process monitoring, design optimization, and uncertainty quantification (Kushwaha et al., 21 Mar 2024, Cruz-González et al., 19 Mar 2025).
Flexible and Modular Multi-Physics Coupling: The modular branch/trunk structure, and the possibility to combine independently pre-trained PI-DeepONets, allows for construction of surrogate solvers for complex coupled systems without retraining from scratch (Goswami et al., 2022).
Limitations: Challenges remain with spectral bias (slow learning of high-frequency content), especially outside the training domain. Recent work on adaptive activations (e.g., Rowdy activation in BubbleONet) addresses this issue for high-frequency regimes (Zhang et al., 5 Aug 2025). Computational bottlenecks posed by collocation-point-based physics loss in high dimensions are actively mitigated via separable architectures and forward-mode autodifferentiation (Mandl et al., 21 Jul 2024, Jiao et al., 7 Jul 2024).

6. Future Research Directions

A range of directions emerge for further development of PI-DeepONet:

Extended Operator Regimes: Research includes extensions to fully nonlinear/nonlocal operators, non-Euclidean and manifold domains, and coupled systems (multi-scale, multi-physics, graph-based PDEs) (Blechschmidt et al., 7 May 2025, Jiao et al., 7 Jul 2024).
Adaptive/Self-Tuning Loss Weights: Effective balancing of physics and data loss terms remains an open field, particularly under uncertainty in data/model (Wang et al., 2021).
Hybrid Approaches and Model Reduction: Synergistic integration of PI-DeepONet learned basis with traditional spectral and reduced-order methods is likely to improve efficiency and interpretability (Williams et al., 27 Nov 2024).
Realistic and Experimental Data Constraint: Increased focus on frameworks for learning from noisy, sparse, or real-world sensor data, possibly combining PI-DeepONet with advanced data assimilation and Bayesian inference for robust prediction and inverse problem solving (Kag et al., 6 Dec 2024, Raj et al., 18 Jan 2025).
Scalability and Extreme-Scale Learning: Separable operator learning architectures, multifidelity surrogates, and modular training/fine-tuning continue to be important to enable applications at scale in science and engineering (Mandl et al., 21 Jul 2024, Jiao et al., 7 Jul 2024, Howard et al., 2022).

7. Resources and Implementation

Open-source code, benchmark datasets, and example implementations for PI-DeepONet and related operator frameworks are available from major groups and repositories, notably:

Documentation often includes training recipes, architecture details, and guidance on applying PI-DeepONet to new classes of PDEs and operator regression problems.

PI-DeepONet thus represents a unifying, robust, and scalable approach for learning nonlinear solution operators under physics constraints, with demonstrated applications across forward modeling, inverse design, parameter estimation, and uncertainty quantification in scientific computing. These features make it an emerging foundation for the next generation of data- and physics-driven modeling in engineering and physical sciences.