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

Kissel, Matthias; Diepold, Klaus

doi:10.1007/978-3-030-46147-8_24

Cited by 2 publications

(2 citation statements)

References 11 publications

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“…As aforementioned, the concept was prior considered in (Czarnecki et al 2017, Kissel and Diepold 2020, Zhu et al 2022. While our contribution can be seen as part of this strategy, we pointed to the novelties, we provide here, complementing this former approaches in section 1.1.…”

Section: Sobolev Trainingmentioning

confidence: 83%

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Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power

Suarez Cardona,

Hecht

2023

Mach. Learn.: Sci. Technol.

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We present novel approximates of variational losses, being applicable for the training of physics-informed neural networks (PINNs). The formulations reflect classic Sobolev space theory for partial differential equations (PDEs) and their weak formulations. The loss approximates rest on polynomial differentiation realised by an extension of classic Gauss–Legendre cubatures, we term Sobolev cubatures, and serve as a replacement of automatic differentiation. We prove the training time complexity of the resulting Sobolev -PINNs with polynomial differentiation to be less than required by PINNs relying on automatic differentiation. On top of one-to-two order of magnitude speed-up the Sobolev-PINNs are demonstrated to achieve closer solution approximations for prominent forward and inverse, linear and non-linear PDE problems compared to established PINNs.

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Section: Sobolev Trainingmentioning

confidence: 83%

“…The derivatives can be computed when the analytical derivative of the ground truth is known. Apart from this rarely appearing case, computations are commonly realised due to automatic differentiation (A.D.) (Baydin et al 2018) or with finite differences (Kissel and Diepold 2020).…”

Section: Contribution-replacing Automatic Differentiation By Polynomi...mentioning

confidence: 99%

Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power

Suarez Cardona,

Hecht

2023

Mach. Learn.: Sci. Technol.

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Machine learning potential for interacting dislocations in the presence of free surfaces

Lanzoni

Rovaris

Montalenti

2022

Sci Rep

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Computing the total energy of a system of N interacting dislocations in the presence of arbitrary free surfaces is a difficult task, requiring Finite Element (FE) numerical calculations. Worst, high accuracy requires very fine meshes in the proximity of each dislocation core. Here we show that FE calculations can be conveniently replaced by a Machine Learning (ML) approach. After formulating the elastic problem in terms of one and two-body terms only, we use Sobolev training to obtain consistent information on both energy and forces, fitted using a feed-forward neural network (NN) architecture. As an example, we apply the proposed methodology to corrugated, heteroepitaxial semiconductor films, searching for the minimum-energy dislocation distributions by using Monte Carlo. Importantly, the presence of an interaction cutoff allows for the application of the method to systems of different sizes without the need to repeat training. Millions of energy evaluations are performed, a task which would have been impossible by brute-force FE calculations. Finally, we show how forces can be exploited in running 2D ML-based dislocation dynamics simulations.

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

Cited by 2 publications

References 11 publications

Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power

Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power

Machine learning potential for interacting dislocations in the presence of free surfaces

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