# Sobolev Training for Neural Networks

@inproceedings{Czarnecki2017SobolevTF, title={Sobolev Training for Neural Networks}, author={Wojciech M. Czarnecki and Simon Osindero and Max Jaderberg and Grzegorz Swirszcz and Razvan Pascanu}, booktitle={NIPS}, year={2017} }

At the heart of deep learning we aim to use neural networks as function approximators - training them to produce outputs from inputs in emulation of a ground truth function or data creation process. [] Key Method By optimising neural networks to not only approximate the function's outputs but also the function's derivatives we encode additional information about the target function within the parameters of the neural network. Thereby we can improve the quality of our predictors, as well as the data…

## 135 Citations

### Sobolev Training for Physics Informed Neural Networks

- Computer Science
- 2021

Inspired by the recent studies that incorporate derivative information for the training of neural networks, a loss function is developed that guides a neural network to reduce the error in the corresponding Sobolev space, making the training substantially efficient.

### Sobolev Training for Implicit Neural Representations with Approximated Image Derivatives

- Computer ScienceECCV
- 2022

This paper proposes a training paradigm for INRs whose target output is image pixels, to encode image derivatives in addition to image values in the neural network, and uses finite differences to approximate image derivatives.

### Sobolev Training with Approximated Derivatives for Black-Box Function Regression with Neural Networks

- Computer ScienceECML/PKDD
- 2019

This paper presents a training pipeline that enables Sobolev Training for regression problems where target derivatives are not directly available and proposes to use a least-squares estimate of the target derivatives based on function values of neighboring training samples.

### JacNet: Learning Functions with Structured Jacobians

- Computer Science, Mathematics
- 2019

This work proposes to directly learn the Jacobian of the input-output function with a neural network, which allows easy control of derivative, and focuses on structuring the derivative to allow invertibility, and also demonstrates other useful priors can be enforced.

### How degenerate is the parametrization of neural networks with the ReLU activation function?

- Computer Science, MathematicsNeurIPS
- 2019

The pathologies which prevent inverse stability in general are presented, and it is shown that by optimizing over such restricted sets, it is still possible to learn any function which can be learned by optimization over unrestricted sets.

### Global Convergence of Sobolev Training for Overparametrized Neural Networks

- Computer ScienceLOD
- 2020

This work proves that an overparameterized two-layer relu neural network trained on the Sobolev loss with gradient flow from random initialization can fit any given function values and any given directional derivatives, under a separation condition on the input data.

### Gradient Regularization Improves Accuracy of Discriminative Models

- Computer ScienceArXiv
- 2017

It is demonstrated through experiments on real and synthetic tasks that stochastic gradient descent is unable to find locally optimal but globally unproductive solutions, and is forced to find solutions that generalize well.

### Neuron Manifold Distillation for Edge Deep Learning

- Computer Science2021 IEEE/ACM 29th International Symposium on Quality of Service (IWQOS)
- 2021

This work proposes a novel neuron manifold distillation (NMD), where the student models not only imitate teacher’s output activations, but also learn the feature geometry structure of the teacher.

### Learning to solve the credit assignment problem

- Computer ScienceICLR
- 2020

A hybrid learning approach that learns to approximate the gradient, and can match or the performance of exact gradient-based learning in both feedforward and convolutional networks.

### Smooth Mathematical Function from Compact Neural Networks

- Computer ScienceArXiv
- 2023

This study gets NNs that generate highly accurate and highly smooth function, which only comprised of a few weight parameters, through discussing a few topics about regression.

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