Physics-Informed Neural Operator for Learning Partial Differential Equations (2111.03794v4)

Abstract: In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, i.e., being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization-convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, e.g., in multi-scale dynamic systems such as Kolmogorov flows.

• The paper presents PINO, which integrates data-driven insights with PDE physics constraints to enhance operator learning accuracy.
• It builds on the Fourier Neural Operator framework to enable zero-shot super-resolution and improve modeling fidelity.
• Numerical experiments on PDEs, including Navier-Stokes, show a 7% error reduction and a 400x speedup over traditional GPU-based solvers.

Overview of "Physics-Informed Neural Operator for Learning Partial Differential Equations"

The paper introduces the concept of Physics-Informed Neural Operators (PINO), an innovative method that integrates data-driven approaches with physics-based constraints to effectively model and solve families of parametric Partial Differential Equations (PDEs). This method addresses key limitations of traditional and data-driven techniques, combining the strength of both realms within a unified framework.

Methodology and Contributions

PINO represents a hybrid approach in solving PDEs, where it leverages both available coarse-resolution data and high-resolution physics-based PDE constraints. This technique is built on the foundation of the Fourier Neural Operator (FNO), a model recognized for its universal approximation capabilities for continuous operators. Unlike standard neural networks that operate on fixed input-output dimensions, neural operators like FNO are adept at mapping entire function spaces. PINO capitalizes on this by incorporating high-resolution PDE constraints which enable it to maintain high fidelity in reconstructing the ground-truth operator.

One of the significant advantages of PINO is its capacity to perform zero-shot super-resolution, accurately predicting highly detailed solutions beyond the resolution of the training data, an area where traditional models often falter. In scenarios where no training data is available, PINO surpasses previous approaches such as Physics-Informed Neural Networks (PINNs), which struggle with optimization issues in multi-scale dynamic systems.

Numerical Results

The paper validates the effectiveness of PINO with several PDEs, including Burgers' Equation, Darcy Flow, and the challenging Navier-Stokes equations in two-dimensional spaces. For instance, PINO achieves significantly lower errors in operator learning tasks when tested on unseen higher-resolution data, outperforming traditional neural operators constrained by limited training data resolutions. PINO demonstrates an approximate 7% reduction in relative error against purely data-trained FNOs in high-resolution Navier-Stokes simulations, achieving a speedup factor of 400x compared to conventional GPU-based solvers.

Implications and Future Directions

PINO's integration of physics-based learning and neural operator approaches sets a precedent for further developments in solving complex PDEs, especially in fields requiring fine-resolution modeling with sparse datasets, such as atmospheric sciences and fluid dynamics. The capability of PINO to generalize without retraining across different scenarios—such as various Reynolds numbers in fluid dynamics—suggests its potential for broad applicability. This might also inspire a new class of hybrid models that synergize data and equation-driven insights to address perennial challenges in scientific computing.

Future research could explore the extension of these methods to three-dimensional PDEs or their application to inverse problems, where recovering input parameters from observed solutions is critical. Furthermore, advancements might involve enhancing the computational efficiency of these models or developing new neural architectures that build on the principles of PINO, offering even greater resolution and accuracy.

In summary, the contributions of this paper lie in the fusion of modeling traditions, pushing the frontier for neural operators with added dimensions of physics-informed learning, thus providing more robust and accurate solutions to complex parametric PDEs.