Physics-Informed DeepONet: Data-Efficient Operator Learning

• PI-DeepONet is an operator learning framework that integrates physical laws via a physics-based loss to overcome data inefficiency and inconsistency in standard DeepONets.
• It employs a composite loss combining mean squared error over paired data and PDE residual penalties via automatic differentiation, achieving high accuracy across benchmarks.
• The framework significantly reduces the need for extensive training data and accelerates inference, making it practical for surrogate modeling in complex parametric PDE problems.

Physics-Informed DeepONet (PI-DeepONet) is an operator learning framework that augments the neural-operator paradigm of DeepONet by incorporating physical laws, typically in the form of partial differential equations (PDEs) and associated boundary or initial conditions, directly into the model training process. By penalizing the violation of governing equations using automatic differentiation within the loss function, PI-DeepONet addresses the data inefficiency and physical inconsistency observed in conventional DeepONets, especially for solving parametric PDEs in scenarios where labeled data is scarce or expensive to acquire.

1. Motivations and Theoretical Foundations

DeepONets are constructed to approximate nonlinear operators $G: u \mapsto s$, with $u$ and $s$ as functions in infinite-dimensional Banach spaces. While DeepONets are theoretically universal approximators for complete operator classes, their practical performance is often limited by the requirement for large sets of paired input-output samples, which may be difficult or costly to obtain. Furthermore, the outputs of data-driven DeepONets can be physically inconsistent—for example, predicted solution derivatives may not align with input fields, directly violating the constraints imposed by the underlying PDEs.

PI-DeepONet addresses these shortcomings by augmenting the learning process with physical regularization, drawing on principles developed in physics-informed neural networks (PINNs). This is operationalized via the addition of a physics-based penalty—derived by automatic differentiation of the neural network output—into the loss function. This constraint biases the learned operator towards outputs that adhere strictly to the governing physical laws, obviating the need for excessive data and supporting generalization to out-of-distribution input conditions (Wang et al., 2021).

2. Methodology: Composite Loss and Automatic Differentiation

The PI-DeepONet framework constructs a composite objective:

$$L(\theta) = L_{\text{operator}}(\theta) + L_{\text{physics}}(\theta)$$

• $L_{\text{operator}}$ is the mean squared error (MSE) loss over any available paired data,

$$L_{\text{operator}}(\theta) = \frac{1}{N P} \sum_{i=1}^N \sum_{j=1}^P \left\| G_{\theta}(u^{(i)})(y_j) - G(u^{(i)})(y_j) \right\|^2$$

• $L_{\text{physics}}$ penalizes the residual of the governing PDE (or ODE), computed using automatic differentiation (AD) of the PI-DeepONet output with respect to input coordinates. For example, for the ODE $ds/dx = u(x)$ with $s(0) = 0$: $$L_{\text{physics}}(\theta) = \frac{1}{N Q} \sum_{i=1}^N \sum_{j=1}^Q \left\| \frac{d}{dx} G_{\theta}(u^{(i)})(x_j) - u^{(i)}(x_j) \right\|^2$$

Automatic differentiation is used to evaluate partial derivatives of $G_\theta(u)(y)$ with respect to spatial and temporal inputs $y$, enabling direct calculation of the PDE residual at arbitrary collocation points. This ensures that the learned operator fits both the data and the physical laws, regularizing model behavior even where training data are absent (Wang et al., 2021).

3. Architectural and Training Considerations

Architecture:

PI-DeepONet retains the standard DeepONet structure, with a branch network encoding the discretized input function at fixed sensor points and a trunk network encoding the evaluation coordinate (e.g., space, time, or both). The output is a bilinear form:

$$G_\theta(u)(y) = \sum_{k=1}^p b_k(u(\xi_1), ..., u(\xi_m)) \cdot t_k(y)$$

where $b_k$ and $t_k$ are the outputs from the branch and trunk networks.

Training:

Model training uses stochastic optimization (e.g., ADAM) and leverages the differentiability of both network output and loss functions via AD. Critically, available paired data (e.g., on boundary or initial conditions) are combined with physical residual loss, and the weighting between $L_{\text{operator}}$ and $L_{\text{physics}}$ can be tuned per problem. This flexibility permits both hybrid (data + physics) and fully physics-driven regimes.

Special Cases:

In situations with no available paired input-output data, PI-DeepONet can be trained solely using physical constraints and a minimal set of initial or boundary conditions, learning the solution operator directly from the PDE structure (Wang et al., 2021).

4. Numerical Results and Performance Assessment

Accuracy and Data Efficiency:

Across benchmark problems—including the anti-derivative operator, diffusion–reaction equations, Burgers’ equation, and the 2D Eikonal equation—PI-DeepONet demonstrates up to one or two orders of magnitude reduction in prediction error compared to standard DeepONets. Notably, when only initial/boundary conditions and no paired data are available, PI-DeepONet can still recover solution operators with high accuracy (Wang et al., 2021).

Reduction in Training Data:

The physics-informed augmentation dramatically reduces the need for large training sets. In some scenarios, the number of required samples drops by up to 100% (i.e., the model can be trained on boundary conditions alone).

Computational Speed-Up:

After training, PI-DeepONet can infer solutions for ${\cal O}(10^3)$ time-dependent PDEs in a fraction of a second, which is up to $10^3\times$ faster than conventional spectral or finite difference solvers.

Generalization:

The physical regularization not only corrects degenerate behavior (such as incorrect derivatives or physical violations) in DeepONet predictions, but also improves robustness to out-of-distribution inputs, as evidenced by accurate extrapolation to unseen initial conditions and parameter regimes.

5. Implications, Extensions, and Potential Limitations

Impact:

PI-DeepONet enables rapid operator-based surrogate modeling in scientific and engineering domains where high-fidelity simulation is expensive. The framework is directly applicable in computational fluid dynamics, heat transfer, materials modeling, computational biology, and Earth science applications. Its ability to enforce physical consistency and operate in the near-absence of labeled data makes it attractive for modeling processes constrained by well-posed PDEs (Wang et al., 2021).

Extensions:

Potential research directions include:

• Exploring optimal neural architectures (e.g., Fourier features, deeper/wider networks) for high-frequency and multi-scale phenomena
• Optimizing composite loss weighting (data vs. physics) for improved model robustness
• Developing advanced optimization techniques to accelerate convergence and overcome gradient pathologies induced by stiff physical constraints
• Scaling the framework to handle multi-physics systems with coupled PDEs.

Limitations:

While PI-DeepONet reduces data requirements and improves generalization, its performance may still be contingent on the expressiveness of the underlying neural networks and the capacity of AD to resolve stiff or irregular operator behavior. Additionally, the weighting of physics vs. data losses remains an open area influencing convergence and solution quality.

6. Data and Code Availability

All data and code supporting the main experimental results, including solution operators for benchmark PDEs, are publicly accessible at:

https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets

This repository enables reproduction of numerical studies, further experimentation, and adaptation of PI-DeepONet to new classes of parametric PDE problems (Wang et al., 2021).

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Physics-informed DeepONet constitutes a significant advance in operator learning for scientific machine learning by integrating physical laws into surrogate model training, yielding physically consistent, data-efficient, and computationally expedient surrogates for a broad range of parametric PDE problems.