Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precision

• URL: http://arxiv.org/abs/2407.17213v1
• Date: Wed, 24 Jul 2024 12:11:09 GMT
• Title: Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precision
• Authors: Jakin Ng, Yongji Wang, Ching-Yao Lai,
• Abstract summary: We propose using the novel multistage neural network approach to learn the residue from the previous stage. We successfully tackle the spectral bias of neural networks. This approach allows the neural network to fit target functions to double floating-point machine precision.
• Score: 1.2663244405597374
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep learning frameworks have become powerful tools for approaching scientific problems such as turbulent flow, which has wide-ranging applications. In practice, however, existing scientific machine learning approaches have difficulty fitting complex, multi-scale dynamical systems to very high precision, as required in scientific contexts. We propose using the novel multistage neural network approach with a spectrum-informed initialization to learn the residue from the previous stage, utilizing the spectral biases associated with neural networks to capture high frequency features in the residue, and successfully tackle the spectral bias of neural networks. This approach allows the neural network to fit target functions to double floating-point machine precision $O(10^{-16})$.

Related papers

• Neural Network Pruning as Spectrum Preserving Process [7.386663473785839]
  We identify the close connection between matrix spectrum learning and neural network training for dense and convolutional layers. We propose a matrix sparsification algorithm tailored for neural network pruning that yields better pruning result.
  arXiv  Detail & Related papers  (2023-07-18T05:39:32Z)
• Multi-stage Neural Networks: Function Approximator of Machine Precision [0.456877715768796]
  We develop multi-stage neural networks that reduce prediction errors below $O(10-16)$ with large network size and extended training iterations. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision $O(10-16)$ of double-floating point within a finite number of iterations.
  arXiv  Detail & Related papers  (2023-07-18T02:47:32Z)
• How neural networks learn to classify chaotic time series [77.34726150561087]
  We study the inner workings of neural networks trained to classify regular-versus-chaotic time series. We find that the relation between input periodicity and activation periodicity is key for the performance of LKCNN models.
  arXiv  Detail & Related papers  (2023-06-04T08:53:27Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Functional Connectome: Approximating Brain Networks with Artificial Neural Networks [1.952097552284465]
  We show that trained deep neural networks are able to capture the computations performed by synthetic biological networks with high accuracy. We show that trained deep neural networks are able to perform zero-shot generalisation in novel environments. Our study reveals a novel and promising direction in systems neuroscience, and can be expanded upon with a multitude of downstream applications.
  arXiv  Detail & Related papers  (2022-11-23T13:12:13Z)
• Precision Machine Learning [5.15188009671301]
  We compare various function approximation methods and study how they scale with increasing parameters and data. We find that neural networks can often outperform classical approximation methods on high-dimensional examples. We develop training tricks which enable us to train neural networks to extremely low loss, close to the limits allowed by numerical precision.
  arXiv  Detail & Related papers  (2022-10-24T17:58:30Z)
• The Spectral Bias of Polynomial Neural Networks [63.27903166253743]
  Polynomial neural networks (PNNs) have been shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a $textitspectral bias$ towards low-frequency functions, which yields faster learning of low-frequency components during training. Inspired by such studies, we conduct a spectral analysis of the Tangent Kernel (NTK) of PNNs. We find that the $Pi$-Net family, i.e., a recently proposed parametrization of PNNs, speeds up the
  arXiv  Detail & Related papers  (2022-02-27T23:12:43Z)
• Dynamic Neural Diversification: Path to Computationally Sustainable Neural Networks [68.8204255655161]
  Small neural networks with a constrained number of trainable parameters, can be suitable resource-efficient candidates for many simple tasks. We explore the diversity of the neurons within the hidden layer during the learning process. We analyze how the diversity of the neurons affects predictions of the model.
  arXiv  Detail & Related papers  (2021-09-20T15:12:16Z)
• A quantum algorithm for training wide and deep classical neural networks [72.2614468437919]
  We show that conditions amenable to classical trainability via gradient descent coincide with those necessary for efficiently solving quantum linear systems. We numerically demonstrate that the MNIST image dataset satisfies such conditions. We provide empirical evidence for $O(log n)$ training of a convolutional neural network with pooling.
  arXiv  Detail & Related papers  (2021-07-19T23:41:03Z)
• Credit Assignment in Neural Networks through Deep Feedback Control [59.14935871979047]
  Deep Feedback Control (DFC) is a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing.
  arXiv  Detail & Related papers  (2021-06-15T05:30:17Z)
• Active Importance Sampling for Variational Objectives Dominated by Rare Events: Consequences for Optimization and Generalization [12.617078020344618]
  We introduce an approach that combines rare events sampling techniques with neural network optimization to optimize objective functions dominated by rare events. We show that importance sampling reduces the variance of the solution to a learning problem, suggesting benefits for generalization. Our numerical experiments demonstrate that we can successfully learn even with the compounding difficulties of high-dimensional and rare data.
  arXiv  Detail & Related papers  (2020-08-11T23:38:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.