A Novel Paradigm in Solving Multiscale Problems

TL;DR

The paper addresses the challenge of simulating multiscale dynamics by introducing a decoupled solving paradigm that treats large-scale dynamics as independent and small-scale dynamics as a slaved system. A Spectral PINN operating in Fourier space learns small-scale behaviour conditioned on the large-scale state, using projections $\mathscr{P}_M$ and $\mathscr{Q}_M$ so that $u = p_M + q_M$ with $p_M \in \mathscr{X}^M$ and $q_M \in (\mathscr{X}^M)^{\perp}$. The authors validate the approach on KS (1D) and NS (2D and 3D) equations, showing clear energy partition between scales and accurate reconstruction of small-scale dynamics via frequency-domain learning, with PDE residuals and initial-condition errors remaining small. They further discuss robustness to non-uniform meshes, complex geometries, noise, and high-dimensional small-scale dynamics, outlining future improvements such as domain decomposition and adaptive meshes. Overall, the work offers a computationally efficient framework that fuses physics-informed learning with spectral representations to tackle multiscale PDEs in fluid dynamics and related fields.

Abstract

Multiscale phenomena manifest across various scientific domains, presenting a ubiquitous challenge in accurately and effectively simulating multiscale dynamics in complex systems. In this paper, a novel decoupling solving paradigm is proposed through modelling large-scale dynamics independently and treating small-scale dynamics as a slaved system. A Spectral Physics-informed Neural Network (PINN) is developed to characterize the small-scale system in an efficient and accurate way, addressing the challenges posed by the representation of multiscale dynamics in neural networks. The effectiveness of the method is demonstrated through extensive numerical experiments, including one-dimensional Kuramot-Sivashinsky equation, two- and three-dimensional Navier-Stokes equations, showcasing its versatility in addressing problems of fluid dynamics. Furthermore, we also delve into the application of the proposed approach to more complex problems, including non-uniform meshes, complex geometries, large-scale data with noise, and high-dimensional small-scale dynamics. The discussions about these scenarios contribute to a comprehensive understanding of the method's capabilities and limitations. By enabling the acquisition of large-scale data with minimal computational demands, coupled with the efficient and accurate characterization of small-scale dynamics via Spectral PINN, our approach offers a valuable and promising approach for researchers seeking to tackle multiscale phenomena effectively.

Figures (7)

• Figure 1: llustrations of the problem set-up and schematic diagram of the spectral PINN configuration. a. The network takes time as input and predicts the Fourier coefficient of the small-scale data, followed by an inverse Fourier Transform. b. Schematic diagram of the spectral PINN method.
• Figure 2: KS equation with the initial condition $u(x,0) = -sin(\pi x/10)$: a. Comprehensive depiction of the full-scale field and extracted large-scale field. b. Comparative analysis of the small-scale dynamics derived from the numerical simulation and the proposed method. c. The training trajectory of the network. d. Comparison of the PDFs of the obtained small-scale fields. e. Comparative assessment of the real and imaginary components of Fourier coefficients signifying the small-scale dynamics, with the mode-specific colouration of data points.
• Figure 3: KS equation with the initial condition $u(x,0) = \cos(x/16)(1 + \sin(x/16))$: a. Comprehensive depiction of the full-scale field and extracted large-scale field. b. Comparative analysis of the small-scale dynamics derived from the numerical simulation and the proposed method. c. The training trajectory of the network. d. Comparison of the PDFs of the obtained small-scale fields. e. Comparative assessment of the real and imaginary components of Fourier coefficients signifying the small-scale dynamics, with mode-specific colouration of data points.
• Figure 4: 2D NS equations: a. A comprehensive depiction of the full-scale, extracted large-scale and small-scale fields derived from the numerical simulation, along with the predicted small-scale dynamics obtained with the proposed method. The detailed results at each time step are provided in the Supplementary Video 1. b. The training trajectory of the network. c. Comparison of the probability density of the obtained small-scale fields. d. Comparative assessment of the real and imaginary components of Fourier coefficients signifying the small-scale dynamics, with mode-specific colouration of data points.
• Figure 5: 3D NS equations: a. Comparative analysis of the small-scale dynamics derived from the numerical simulation and the proposed method. The detailed results at each time step are provided in the Supplementary Video 2. b. Comparative assessment of the real and imaginary components of Fourier coefficients signifying the small-scale dynamics, with mode-specific colouration of data points. c. The training trajectory of the network. d. Comparison of the probability density of the obtained small-scale fields.

Theorems & Definitions (2)

• Remark 1
• Remark 2