Geometry-Informed Neural Operator for Large-Scale 3D PDEs (2309.00583v1)

Abstract: We propose the geometry-informed neural operator (GINO), a highly efficient approach to learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function and point-cloud representations of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. GINO is discretization-convergent, meaning the trained model can be applied to arbitrary discretization of the continuous domain and it converges to the continuum operator as the discretization is refined. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a $26,000 \times$ speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

• The paper introduces GINO, a novel framework combining graph and Fourier neural operators to accelerate and improve the accuracy of large-scale 3D PDE simulations.
• It employs geometry encoders and decoders with signed distance functions and point-cloud representations to handle irregular meshes and complex geometries.
• Experiments reveal a 26,000× speed-up and a 25% error reduction over traditional methods, demonstrating its transformative impact on computational fluid dynamics.

Analysis of "Geometry-Informed Neural Operator for Large-Scale 3D PDEs"

The paper "Geometry-Informed Neural Operator for Large-Scale 3D PDEs" proposes an innovative framework, termed the Geometry-Informed Neural Operator (GINO), designed to efficiently learn solution operators for partial differential equations (PDEs) with varying geometries. It tackles the computational demands faced in scientific computing, particularly within the field of computational fluid dynamics (CFD) simulations, by merging graph neural operators (GNO) and Fourier neural operators (FNO) into a unified architecture.

One of the notable aspects of GINO is its ability to process large-scale 3D PDEs across diverse geometrical configurations. This is facilitated through the incorporation of signed distance functions (SDF) and point-cloud representations, which enable the model to accurately handle the complexities associated with 3D geometries. The architecture leverages the locality of GNOs for input-output mappings and the global coverage of FNOs to manage the extensive integration operations often required in solving PDEs.

Methodology

The GINO architecture is structured into three core components:

1. Geometry Encoder: This segment employs GNOs to manage irregular grids, transforming input point clouds into latent space representations. This encoding is crucial for handling different geometries and irregular mesh configurations effectively.
2. Global Model: FNOs are employed to capture global interactions efficiently. By transitioning computations to a regular grid space, GINO benefits from the computational efficiency of Fast Fourier Transforms (FFT), thus significantly speeding up the simulation processes.
3. Geometry Decoder: The final output predictions, such as pressure or velocity fields, are retrieved in the original geometry space using GNOs, supporting arbitrary point evaluations for increased flexibility.

The model demonstrates remarkable performance in experimental evaluations with the industry-standard aerodynamics dataset, demonstrating an astonishing speed-up of $26,000\times$ over traditional GPU-optimized CFD solvers. This efficiency is coupled with a substantial reduction in error rates when predicting outcomes for unseen configurations and conditions.

Numerical Results and Implications

Empirical testing highlights GINO’s ability to reduce error rates by one-fourth when compared to deep neural network approaches. Furthermore, when investigating the cost-accuracy trade-offs, GINO demonstrates a transformative potential in computational efficiency. Its discretization-convergent design denotes that as the resolution of input discretization is increased, the approximated solution approaches the true solution. This property is vital for ensuring the reliability of simulations across varying conditions and resolutions.

The implications of this work are multifaceted. Practically, GINO can streamline CFD processes, potentially revolutionizing design cycles in automotive and aerospace sectors by reducing computational time and cost. Theoretically, the integration of GNOs and FNOs within a single framework opens new avenues for exploring continuous function space mappings in neural networks.

Future Outlook

Future research could extend the application of GINO across other domains where large-scale PDEs play a crucial role, such as weather forecasting and reservoir engineering. Further exploration into the scalability of GINO, particularly in higher-dimensional problem spaces, could unveil additional insights into its applicability and robustness.

In summary, this paper contributes a significant methodological advancement in the field of operator learning for PDEs. The GINO framework, with its nuanced approach to geometry-informed prediction and computational efficiency, offers promising potential for accelerating and enhancing simulations in numerous scientific and engineering applications.