LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE

TL;DR

LaPON is a hybrid framework that combines neural operators with traditional numerical methods, where neural operators are used to compensate for the effect of discretization errors on the analytical scale in under-resolution simulations, showing the potential for wide application in fields such as weather forecasting and engineering design.

Abstract

Accelerating the solution of nonlinear partial differential equations (PDEs) while maintaining accuracy at coarse spatiotemporal resolution remains a key challenge in scientific computing. Physics-informed machine learning (ML) methods such as Physics-Informed Neural Networks (PINNs) introduce prior knowledge through loss functions to ensure physical consistency, but their "soft constraints" are usually not strictly satisfied. Here, we propose LaPON, an operator network inspired by the Lagrange's mean value theorem, which embeds prior knowledge directly into the neural network architecture instead of the loss function, making the neural network naturally satisfy the given constraints. This is a hybrid framework that combines neural operators with traditional numerical methods, where neural operators are used to compensate for the effect of discretization errors on the analytical scale in under-resolution simulations. As evaluated on turbulence problem modeled by the Navier-Stokes equations (NSE), the multiple time step extrapolation accuracy and stability of LaPON exceed the direct numerical simulation baseline at 8x coarser grids and 8x larger time steps, while achieving a vorticity correlation of more than 0.98 with the ground truth. It is worth noting that the model can be well generalized to unseen flow states, such as turbulence with different forcing, without retraining. In addition, with the same training data, LaPON's comprehensive metrics on the out-of-distribution test set are at least approximately twice as good as two popular ML baseline methods. By combining numerical computing with machine learning, LaPON provides a scalable and reliable solution for high-fidelity fluid dynamics simulation, showing the potential for wide application in fields such as weather forecasting and engineering design.

Figures (5)

• Figure 1: Overview of our approach. (a) Illustrative examples of training and validation, showing the strong generalization capabilities of our model. (b) Structure of a single time step for our model, which integrates numerical computation and AI correction operators implemented by the convolutional neural networks. (c) Examples of isotropic convolution kernels of different sizes (in the solid box) and the corresponding rotational symmetry.
• Figure 2: LaPON exceeds accuracy of direct simulation at $8 \times$ higher spatial resolution, and the accuracy is also significantly higher on large time steps. (a) The vorticity field solved by our model (LaPON $64 \times 64$) and baseline (DS $64 \times 64$) on $8 \times$ and $21 \times$ base time step, and the corresponding error. (b) Comparison of the vorticity correlation between predicted flows and the reference solution for our model and direct simulation on different time step sizes. (c) Comparison of the vorticity correlation between predicted flows and the reference solution for our model and direct simulation on different spatial resolutions. (d) Energy spectrum on the first one Kolmogorov time (210 original time steps) using $1 \times$ the base time step. The gray vertical line indicates the effective wavenumber cutoff region of $64 \times 64$ resolution.
• Figure 3: LaPON maintains accuracy and stability and exceeds direct simulation at $8 \times$ higher spatiotemporal resolution on large time steps that do not even satisfy the CFL condition. (a) Time evolution of the vorticity field (using $1 \times$ the base time step) and the corresponding reference solution. (b) Comparison of the mean and standard deviation (in the sample dimension) of the vorticity correlation between the prediction of our model and the direct simulation baselines and the reference solution. The gray vertical solid line represents the first one Kolmogorov time. (c) Comparison of the noise immunity for our model and direct simulation baselines on different noise level (using $8 \times$ the base time step). (d) Compare the robustness of our model to the direct simulation baseline against partial data missing (using $8 \times$ the base time step). (e) Visualization of data noise and local missing.
• Figure 4: LaPON generalizes well to decaying turbulence without retraining. (a) Prediction and error of vorticity fields. (b) Vorticity correlation between predicted flows and the reference solution. (c) Energy spectrum on the first 210 original time steps using $1 \times$ the base time step.
• Figure 5: LaPON outperform two classic ML baseline methods in terms of accuracy, stability and generalization (the evolution time used in the test is about 210 original time steps). (a) The mean and standard deviation of vorticity correlation for each architecture on the training dataset using $8 \times$ the base time step and $64 \times 64$ spatial resolution. (b) Each row within a subplot shows the performance metric (1% low value of vorticity correlation) of one architecture. The models are trained on isotropic forced turbulence with a certain operating parameter (same as Section \ref{['sec:eval_acc']}) and tested for generalization without retraining on four types of out-of-distribution data (extra time, extra space domain, decaying and more turbulent flows with different fluid properties and forcing types). The three columns of subplots correspond to three different spatial and temporal discrete scales ($\Delta t$ is the base time step, $\Delta x_i$ is the grid size of the original dataset). Under the same training data and configuration, the composite metrics of LaPON on the test set are at least about 1 times better than the two popular ML baseline methods.