# A Novel Paradigm in Solving Multiscale Problems

arXiv: https://arxiv.org/abs/2402.05067 · HTML: https://mlsanity.com/paper/2402.05067/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 56 references that are in the map.
- First posted on arXiv 7 Feb 2024.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [A Closer Look at Deep Learning Heuristics: Learning rate restarts, Warmup and Distillation](https://mlsanity.com/paper/1810.13243/) (2018)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Scheduling Techniques for Liver Segmentation: ReduceLRonPlateau Vs OneCycleLR](https://mlsanity.com/paper/2202.06373/) (2022)

## Abstract (arXiv)

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.