PI-DeepONet: Physics-Informed Deep Operator Networks

• PI-DeepONet is a physics-informed deep operator network that integrates PDE constraints into training to ensure physical consistency and significantly reduce data dependence.
• It employs automatic differentiation and physics-based penalty terms, achieving orders of magnitude reduction in relative errors for benchmark PDEs.
• The approach accelerates inference across large families of PDEs, enabling rapid real-time simulations and efficient design optimization in complex systems.

Physics-Informed Deep Operator Networks (PI-DeepONet) are a model class that extend deep operator networks for learning parametric solution operators of partial differential equations (PDEs) by embedding the governing physical laws directly into the training procedure. This hybrid approach substantially improves both physical consistency in the learned operator and data efficiency, enabling accurate predictions even in the data-scarce regime and accelerating solution generation across large families of PDE instances (Wang et al., 2021). The PI-DeepONet framework achieves these properties by augmenting the standard DeepONet architecture with automatic differentiation and physics-based loss penalization, biasing the learned operator to satisfy the underlying differential equations.

1. Foundations and Motivation

PI-DeepONet addresses two major deficiencies in traditional operator learning: (1) dependence on large paired datasets of input–output functions, and (2) lack of guarantees that learned outputs satisfy the governing laws of physics embodied in PDE models. Classical DeepONets, although theoretically capable of approximating general nonlinear operators between infinite-dimensional Banach spaces, often produce solutions that violate conservation, boundary, or evolutionary properties encoded in the true PDE (Wang et al., 2021). Generating sufficient and reliable paired data for operator learning is itself expensive or infeasible for many physical systems. By integrating the PDEs via soft penalty constraints (physics-informed regularization) into the loss function and leveraging automatic differentiation for fast, accurate derivative computation, PI-DeepONet rectifies these issues, enabling both data efficiency and physical reliability.

2. Network Architecture and Physics-Informed Loss

The standard DeepONet architecture consists of a branch network that encodes the input function (typically by sampling at prescribed sensor points) and a trunk network that encodes the output coordinates (e.g., points in time/space). The final prediction is produced by taking an inner product between branch and trunk outputs: $$\mathcal{G}_\theta(u)(y) = \sum_{k} b_k(u(x_1), \ldots, u(x_m)) \, t_k(y)$$ where $b_k$ are outputs of the branch net (input function encoding), and $t_k$ are outputs of the trunk net (coordinate encoding).

PI-DeepONet augments the training loss by introducing a physics term in addition to any available data mismatch loss: $$\mathcal{L}(\theta) = \mathcal{L}_\text{operator}(\theta) + \mathcal{L}_\text{physics}(\theta)$$

• $\mathcal{L}_\text{operator}$: Standard mean squared error between network outputs and available “ground truth” data or known constraints (e.g., initial/boundary conditions).
• $\mathcal{L}_\text{physics}$: Penalty enforcing that the network output satisfies the relevant PDE at a set of collocation points, computed as the mean squared residual of the PDE’s differential operator, typically via automatic differentiation.

For example, for a network representation $\hat{u}_\theta(x, t)$, physics loss for a generic PDE $\mathcal{N}(u) = 0$ is: $$\mathcal{L}_\text{physics}(\theta) = \frac{1}{N}\sum_{i=1}^{N} |\mathcal{N}(\hat{u}_\theta)(x_i, t_i)|^2$$ This enables robust training even without explicit paired input–output data, as long as the PDE structure and initial/boundary conditions are known.

3. Performance Improvements and Computational Advantages

PI-DeepONet yields marked improvements in predictive accuracy, especially in data-scarce regimes. Quantitatively, relative $L_2$ errors on benchmark PDEs (such as nonlinear diffusion-reaction and Burgers’ equations) are reduced by up to one or two orders of magnitude when compared to standard DeepONet (Wang et al., 2021). In some cases, the PI-DeepONet can be trained to high accuracy using only boundary or initial conditions with no paired output at all.

A striking computational benefit of the operator-learning setting is rapid inference: a single trained PI-DeepONet can evaluate the solution operator for $\mathcal{O}(10^3)$ distinct PDE instances in a fraction of a second—three orders of magnitude faster than high-fidelity spectral or finite-difference solvers. This efficiency is particularly attractive for applications in real-time modeling, uncertainty quantification, or iterative design/optimization loops.

4. Empirical Evidence and Application Domains

Extensive numerical studies demonstrate the flexibility and generality of PI-DeepONet. Examples in (Wang et al., 2021) include:

• Anti-derivative operator: The methods enforce exact correspondence between the predicted solution and the input function through its spatial derivative, outperforming data-driven DeepONet with limited data.
• Nonlinear diffusion-reaction PDEs: Relative error reduction from 1.92% (DeepONet) to 0.45% (PI-DeepONet).
• Burgers’ equation: Physics loss weight tuning and architectural modifications drive error down from 17% (unmodified) to as low as 1.38%.
• Eikonal equation: Solution operator maps from geometry parameters (or signed distance function) to the solution, with observed relative $L_2$ errors near $4.22 \times 10^{-3}$.

Irrespective of the underlying PDE family and input dimensionality, PI-DeepONet produces physically consistent solution maps that inherit the modeled system’s qualitative behavior beyond direct training data.

5. Implementation Considerations

Key implementation aspects for PI-DeepONet include:

Aspect	Recommendation / Constraint	Potential Limitation
Architecture	DeepONet with two sub-networks	Custom architectures for multi-scale phenomena are beneficial
Derivative computation	Leverage automatic differentiation	Must ensure continuous differentiability in NN outputs
Loss balancing	Careful weighting of data vs. physics loss	Manual or meta-learned tuning required
Input encoding	Input functions sampled at sensor points	Sensor placement can affect accuracy
Collocation strategy	Random or structured sampling in domain	Impacts enforcement of physics constraints

For challenging systems, adapting the network structure (e.g., using Fourier feature networks, increasing width/depth, or introducing physics-informed architectural biases) may be required to resolve high-frequency or multi-scale solution features.

6. Practical Impact and Future Directions

The PI-DeepONet paradigm is broadly applicable in scientific and engineering domains, particularly where simulation speed and data efficiency are paramount:

• Real-time simulation and control (fluid/thermal systems): Rapid, physically faithful surrogate models replace online CFD/FEA solvers.
• Design optimization: Operator surrogates enable thousands of forward evaluations required for design-of-experiments and sensitivity analysis.
• Uncertainty quantification & inverse problems: Physical constraints enhance identifiability and stabilize learning under sparse or noisy data.
• Shape-parameterized PDEs: As seen in Eikonal and signed distance function experiments, operator learning can capture solution dependence on variable domains.

Suggested avenues for further research include: optimal architecture adaptation for multi-scale or oscillatory behavior, systematic loss weight selection, scalable training strategies (for extremely high-dimensional domains), and coupling with meta-learning or automated architecture search. Extending the framework to integrate with other operator learning paradigms, or to accommodate problems involving coupled or hybrid-physics PDEs, remains an open and fertile direction.

7. Summary Table: PI-DeepONet Properties

Property	PI-DeepONet	DeepONet (Standard)
Data requirement	Low—boundary/initial conditions suffice	High—paired input–output
Physical consistency	Enforced via PDE constraints	Not guaranteed
Predictive accuracy	High—robust to data scarcity	Data-dependent
Speed of inference	Orders of magnitude faster than classical solvers	Orders faster than data-driven solvers, physical violation possible
Breadth of application	Parametric, nonlinear, time-dependent PDEs	General operator learning, not physics-aware

PI-DeepONet, by embedding physical laws within a universal operator learning framework, achieves a principled synthesis of scientific computing and neural approximation, combining the data-efficiency and interpretability of PDE-based models with the flexibility of deep learning (Wang et al., 2021).