Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

Han, Jiequn; Lu, Jianfeng; Zhou, Mo

doi:10.1016/j.jcp.2020.109792

Cited by 68 publications

(54 citation statements)

References 11 publications

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“…When developing machine learning models (such as ANNs), at a certain point, more features or dimensions can decrease model's accuracy (a phenomenon called ‘the curse of dimensionality’) ( Han et al, 2020 ). In this case, a dimensionality reduction strategy (such as the utilization of PCA) may be employed in order to accelerate the training time of the model, reduce its complexity and avoid overfitting ( Becker et al, 2020 ).…”

Section: Methodsmentioning

confidence: 99%

Determination of the physical state of a drug in amorphous solid dispersions using artificial neural networks and ATR-FTIR spectroscopy

Kapourani

Valkanioti

Kontogiannopoulos

et al. 2020

International Journal of Pharmaceutics: X

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The objective of the present study was to evaluate the use of artificial neural networks (ANNs) in the development of a new chemometric model that will be able to simultaneously distinguish and quantify the percentage of the crystalline and the neat amorphous drug located within the drug-rich amorphous zones formed in an amorphous solid dispersion (ASD) system. Attenuated total reflectance Fourier-transform infrared (ATR-FTIR) spectroscopy was used, while Rivaroxaban (RIV, drug) and Soluplus® (SOL, matrix-carrier) were selected for the preparation of a suitable ASD model system. Adequate calibration and test sets were prepared by spiking different percentages of the crystalline and the amorphous drug in the ASDs (prepared by the melting - quench cooling approach), while a 2 4 full factorial experimental design was employed for the screening of ANN's structure and training parameters as well as spectra region selection and data preprocessing. Results showed increased prediction performance, measured based on the root mean squared error of prediction (RMSEp) for the test sample, for both the crystalline (RMSEp (crystal) = 0.86) and the amorphous (RMSEp (amorphous) = 2.14) drug. Comparison with traditional regression techniques, such as partial least square and principle component regressions, revealed the superiority of ANNs, indicating that in cases of high structural similarity between the investigated compounds (i.e., the crystalline and the amorphous forms of the same compound) the implementation of more powerful/sophisticated regression techniques, such as ANNs, is mandatory.

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Section: Methodsmentioning

confidence: 99%

Determination of the physical state of a drug in amorphous solid dispersions using artificial neural networks and ATR-FTIR spectroscopy

Kapourani

Valkanioti

Kontogiannopoulos

et al. 2020

International Journal of Pharmaceutics: X

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“…Based on the natural idea of representing solutions of PDEs by (deep) neural networks, different loss functions for solving PDEs are proposed. [26,27] utilize the Feynman-Kac formulation which turns solving PDE to a stochastic control problem and the weak adversarial network [81] solves the weak formulations of PDEs via an adversarial network.…”

Section: Related Workmentioning

confidence: 99%

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

Lu¹,

Chen²,

Lu³

et al. 2021

Preprint

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In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To simplify the problem, we focus on a prototype elliptic PDE: the Schrödinger equation on a hypercube with zero Dirichlet boundary condition, which is applied in quantum-mechanical systems. We establish upper and lower bounds for both methods, which improve upon concurrently developed upper bounds for this problem via a fast rate generalization bound. We discover that the current Deep Ritz Method is sub-optimal and propose a modified version of it. We also prove that PINN and the modified version of DRM can achieve minimax optimal bounds over Sobolev spaces. Empirically, following recent work which has shown that the deep model accuracy will improve with growing training sets according to a power law, we supply computational experiments to show similar-behavior of dimension dependent power law for deep PDE solvers.

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“…Motivated by those recent success, researchers have been actively exploring using deep learning techniques to solve high dimensional PDEs [10,18,22,23,29,37,45,48] by using neural networks to parameterize the unknown solution of high dimensional PDEs. Thanks to the flexibility of the neural network approximations, such methods have achieved remarkable results for various kind of PDE problems, including eigenvalue problems for many-body quantum systems (see e.g., [7,9,12,19,[24][25][26]36]), where the high dimensional wave functions are parameterized by neural networks with specific architecture design to address the symmetry properties of many-body quantum systems.…”

Section: Introductionmentioning

confidence: 99%

A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems

2022

Comm. Amer. Math. Soc.

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This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a d d -dimensional hypercube with Neumann boundary condition. We prove that the convergence rate of the generalization error is independent of dimension d d , under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The latter is achieved by a fixed point argument based on the Krein-Rutman theorem.

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Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

Cited by 68 publications

References 11 publications

Determination of the physical state of a drug in amorphous solid dispersions using artificial neural networks and ATR-FTIR spectroscopy

Determination of the physical state of a drug in amorphous solid dispersions using artificial neural networks and ATR-FTIR spectroscopy

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems

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