Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs

Xi-An Li; Zhi-Qin John Xu; Lei Zhang

Home/Archive/arXiv

arXiv · 2112.06660

Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs

Xi-An Li·Zhi-Qin John Xu·Lei Zhang

View original source ↗Read source document ↗Export metadata ↓

Abstract

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the "frequency principle" that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD$^2$NN) architecture for a class of multi-scale problems by combining traditional numerical analysis ideas and MscaleDNN algorithms. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions, respectively. In addition, a novel trigonometric activation function is incorporated in the SD$^2$NN model. We demonstrate the performance of the SD$^2$NN architecture through several benchmark multi-scale problems in regular or irregular geometric domains. Numerical results show that the SD$^2$NN model is superior to existing models such as MscaleDNN.

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Xi-An Li, Zhi-Qin John Xu, Lei Zhang. 2021-12-16. Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs. https://arxiv.org/abs/2112.06660

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Online Regularized Statistical Learning in Reproducing Kernel Hilbert Space With Non-Stationary Data

We study recursive regularized learning algorithms in the reproducing kernel Hilbert space (RKHS) with non-stationary online data streams. We introduce the concept of a random Tikhonov regularization path and decompose the tracking error of the algorithm's output for the regularization path into random difference equations in RKHS. We show that the tracking error vanishes in mean square and almost surely if the regularization path is slowly time-varying. Then, leveraging the monotonicity of inverse operators and the spectral decomposition of compact operators, and introducing the RKHS persistence of excitation condition, we develop a dominated convergence method to prove the mean square and almost sure consistency between the regularization path and the unknown function to be learned. Especially, for independent and non-identically distributed data streams, the mean square and almost sure consistency between the algorithm's output and the unknown function is achieved if the input data's marginal probability measures are slowly time-varying and the average measure over each fixed-length time period is uniformly above a strictly positive finite Borel measure.

cs.LG↗

arXiv2026

Reflective Policy Optimization

On-policy reinforcement learning methods, like Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO), often demand extensive data per update, leading to sample inefficiency. This paper introduces Reflective Policy Optimization (RPO), a novel on-policy extension that amalgamates past and future state-action information for policy optimization. This approach empowers the agent for introspection, allowing modifications to its actions within the current state. Theoretical analysis confirms that policy performance is monotonically improved and contracts the solution space, consequently expediting the convergence procedure. Empirical results demonstrate RPO's feasibility and efficacy in two reinforcement learning benchmarks, culminating in superior sample efficiency. The source code of this work is available at https://github.com/Edgargan/RPO.

cs.LG↗

arXiv2026

Transductive Off-policy Proximal Policy Optimization

Proximal Policy Optimization (PPO) is a popular model-free reinforcement learning algorithm, esteemed for its simplicity and efficacy. However, due to its inherent on-policy nature, its proficiency in harnessing data from disparate policies is constrained. This paper introduces a novel off-policy extension to the original PPO method, christened Transductive Off-policy PPO (ToPPO). Herein, we provide theoretical justification for incorporating off-policy data in PPO training and prudent guidelines for its safe application. Our contribution includes a novel formulation of the policy improvement lower bound for prospective policies derived from off-policy data, accompanied by a computationally efficient mechanism to optimize this bound, underpinned by assurances of monotonic improvement. Comprehensive experimental results across six representative tasks underscore ToPPO's promising performance.

cs.LG↗