A Quasilinear Algorithm for Computing Higher-Order Derivatives of Deep Feed-Forward Neural Networks
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909425734778880 |
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| author | Chickering, Kyle R. |
| author_facet | Chickering, Kyle R. |
| contents | The use of neural networks for solving differential equations is practically difficult due to the exponentially increasing runtime of autodifferentiation when computing high-order derivatives. We propose $n$-TangentProp, the natural extension of the TangentProp formalism \cite{simard1991tangent} to arbitrarily many derivatives. $n$-TangentProp computes the exact derivative $d^n/dx^n f(x)$ in quasilinear, instead of exponential time, for a densely connected, feed-forward neural network $f$ with a smooth, parameter-free activation function. We validate our algorithm empirically across a range of depths, widths, and number of derivatives. We demonstrate that our method is particularly beneficial in the context of physics-informed neural networks where \ntp allows for significantly faster training times than previous methods and has favorable scaling with respect to both model size and loss-function complexity as measured by the number of required derivatives. The code for this paper can be found at https://github.com/kyrochi/n\_tangentprop. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Quasilinear Algorithm for Computing Higher-Order Derivatives of Deep Feed-Forward Neural Networks Chickering, Kyle R. Machine Learning The use of neural networks for solving differential equations is practically difficult due to the exponentially increasing runtime of autodifferentiation when computing high-order derivatives. We propose $n$-TangentProp, the natural extension of the TangentProp formalism \cite{simard1991tangent} to arbitrarily many derivatives. $n$-TangentProp computes the exact derivative $d^n/dx^n f(x)$ in quasilinear, instead of exponential time, for a densely connected, feed-forward neural network $f$ with a smooth, parameter-free activation function. We validate our algorithm empirically across a range of depths, widths, and number of derivatives. We demonstrate that our method is particularly beneficial in the context of physics-informed neural networks where \ntp allows for significantly faster training times than previous methods and has favorable scaling with respect to both model size and loss-function complexity as measured by the number of required derivatives. The code for this paper can be found at https://github.com/kyrochi/n\_tangentprop. |
| title | A Quasilinear Algorithm for Computing Higher-Order Derivatives of Deep Feed-Forward Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2412.09752 |