# Frequency-adaptive tensor neural networks for high-dimensional multi-scale problems

arXiv: https://arxiv.org/abs/2508.15198 · HTML: https://mlsanity.com/paper/2508.15198/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 21 Aug 2025.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [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)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Reconstructing continuous distributions of 3D protein structure from cryo-EM images](https://mlsanity.com/paper/1909.05215/) (2019)
- [Understanding training and generalization in deep learning by Fourier analysis](https://mlsanity.com/paper/1808.04295/) (2018)
- [A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems](https://mlsanity.com/paper/1909.11759/) (2019)
- [Multi-scale Deep Neural Networks for Solving High Dimensional PDEs](https://mlsanity.com/paper/1910.11710/) (2019)
- [Explicitizing an Implicit Bias of the Frequency Principle in Two-layer Neural Networks](https://mlsanity.com/paper/1905.10264/) (2019)
- [Reproducing Activation Function for Deep Learning](https://mlsanity.com/paper/2101.04844/) (2021)
- [Functional Tensor Decompositions for Physics-Informed Neural Networks](https://mlsanity.com/paper/2408.13101/) (2024)
- [Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs](https://mlsanity.com/paper/2112.06660/) (2021)
- [Frequency-adaptive Multi-scale Deep Neural Networks](https://mlsanity.com/paper/2410.00053/) (2024)

## Abstract (arXiv)

Tensor neural networks (TNNs) have demonstrated their superiority in solving high-dimensional problems. However, similar to conventional neural networks, TNNs are also influenced by the Frequency Principle, which limits their ability to accurately capture high-frequency features of the solution. In this work, we analyze the training dynamics of TNNs by Fourier analysis and enhance their expressivity for high-dimensional multi-scale problems by incorporating random Fourier features. Leveraging the inherent tensor structure of TNNs, we further propose a novel approach to extract frequency features of high-dimensional functions by performing the Discrete Fourier Transform to one-dimensional component functions. This strategy effectively mitigates the curse of dimensionality. Building on this idea, we propose a frequency-adaptive TNNs algorithm, which significantly improves the ability of TNNs in solving complex multi-scale problems. Extensive numerical experiments are performed to validate the effectiveness and robustness of the proposed frequency-adaptive TNNs algorithm.