DeepMoD: Deep learning for Model Discovery in noisy data
DeepMoD is a deep learning based model discovery algorithm which seeks the partial differential equation underlying a spatio-temporal data set. DeepMoD employs sparse regression on a library of basis functions and their corresponding spatial derivatives. A feed-forward neural network approximates the data set and automatic differentiation is used to construct this function library and perform regression within the neural network. This construction makes it extremely robust to noise and applicable to small data sets and, contrary to other deep learning methods, does not require a training set and is impervious to overfitting. We illustrate this approach on several physical problems, such as the Burgers', Korteweg-de Vries, advection-diffusion and Keller-Segel equations, and find that it requires as few as O(10^2) samples and works at noise levels up to 75%. This resilience to noise and high performance at very few samples highlights the potential of this method to be applied on experimental data. Code and examples available at https://github.com/PhIMaL/DeePyMoD.
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Reference:
DeepMoD: Deep learning for Model Discovery in noisy data ; G.-J. Both, S. Choudhury, P. Sens and R. Kusters; arXiv (REF)
Physics Inspiered Machine Learning of the Cytoskeleton
The interior of a cell is a highly complex and viscoelastic environment, responding to growth and motion differently at various length and timescales. This leads to difficulties in experimentally validating mechanical models of the cellular cytoskeleton. The goal of this project is to apply a Physics Informed Neural Network [1,2] or sparse regression [3] algorithms to identify (potentially nonlinear) relationships between physical quantities from possibly high-dimensional experimental data which describe the spatiotemporal dynamics of the cytoskeleton. The use of neural networks and novel inference algorithms can extract previously inaccessible quantities from experimental data. In physics informed neural networks, principle physical laws that govern the time-dependent dynamics of the system are implemented as prior information in the algorithm, constraining the space of admissible solutions to a manageable size. In the case of a cytoskeletal network, such prior information can be the constitutive relations or conservation laws.
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References:
[1] Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations; Raissi M, Perdikaris P, Karniadakis GE.; arXiv:1711.10561 (2017)
[2] Physics informed deep learning (Part II): data-driven discovery of nonlinear partial differential equations; Raissi M, Perdikaris P, Karniadakis GE.; arXiv:1711.10566 (2017)
[3] Data-driven discovery of partial differential equations; S. H. Rudy, S. L. Brunton, J. L. Proctor, N. Kutz.; Science Advances, 3(4), e1602614 (2017)