# AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems

arXiv: https://arxiv.org/abs/2207.11735 · HTML: https://mlsanity.com/paper/2207.11735/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 24 Jul 2022.

## Builds on (most cited first, 10 of 10)

- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)
- [Accelerating Physics-Informed Neural Network Training with Prior Dictionaries](https://mlsanity.com/paper/2004.08151/) (2020)
- [Deep Multiscale Model Learning](https://mlsanity.com/paper/1806.04830/) (2018)
- [Variational training of neural network approximations of solution maps for physical models](https://mlsanity.com/paper/1905.02789/) (2019)
- [Bayesian Sparse learning with preconditioned stochastic gradient MCMC and its applications](https://mlsanity.com/paper/2006.16376/) (2020)
- [A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction](https://mlsanity.com/paper/1907.00806/) (2019)

## Built on by (most cited first, 2 of 2)

- [SENDAI: A Hierarchical Sparse-measurement, EfficieNt Data AssImilation Framework](https://mlsanity.com/paper/2601.21664/) (2026)
- [Gaussian-process surrogate indicators for residual-based adaptive GMsFEM](https://mlsanity.com/paper/2609.26843/) (2026)

## Abstract (arXiv)

In this work, we propose an adaptive sparse learning algorithm that can be applied to learn the physical processes and obtain a sparse representation of the solution given a large snapshot space. Assume that there is a rich class of precomputed basis functions that can be used to approximate the quantity of interest. We then design a neural network architecture to learn the coefficients of solutions in the spaces which are spanned by these basis functions. The information of the basis functions are incorporated in the loss function, which minimizes the differences between the downscaled reduced order solutions and reference solutions at multiple time steps. The network contains multiple submodules and the solutions at different time steps can be learned simultaneously. We propose some strategies in the learning framework to identify important degrees of freedom. To find a sparse solution representation, a soft thresholding operator is applied to enforce the sparsity of the output coefficient vectors of the neural network. To avoid over-simplification and enrich the approximation space, some degrees of freedom can be added back to the system through a greedy algorithm. In both scenarios, that is, removing and adding degrees of freedom, the corresponding network connections are pruned or reactivated guided by the magnitude of the solution coefficients obtained from the network outputs. The proposed adaptive learning process is applied to some toy case examples to demonstrate that it can achieve a good basis selection and accurate approximation. More numerical tests are performed on two-phase multiscale flow problems to show the capability and interpretability of the proposed method on complicated applications.