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Rotation Equivariant Operators for Machine Learning on Scalar and Vector Fields
[article]
2022
arXiv
pre-print
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We implement the Julia package EquivariantOperators.jl for fully differentiable finite difference equivariant operators on scalar, vector and higher order tensor fields in 2d/3d. ...
It can run forwards for simulation or image processing, or be back propagated for computer vision, inverse problems and optimal control. ...
Differentiating against control parameters or forcing functions enables solving optimal control problems. ...
arXiv:2108.09541v3
fatcat:msszgupzczgpddkqike2pqoa2q
Scientific Machine Learning for Modeling and Simulating Complex Fluids
[article]
2022
arXiv
pre-print
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By construction, these models respect physical constraints, such as frame-invariance and tensor symmetry, required by continuum mechanics. ...
Consequently, early machine learning constitutive equations have not been portable between different deformation protocols or mechanical observables. ...
Materials and Methods Tensor Basis Neural Network The Tensor Basis Neural Network employed in this work is a linear basis expansion presented in equation 2. ...
arXiv:2210.04431v1
fatcat:jlvi76zz7rb3lfunyyocflctxi
Physics-driven machine learning models coupling PyTorch and Firedrake
[article]
2023
arXiv
pre-print
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2023 pre-print arXiv:2303.06871v2 fatcat:fii4bu2wpjhiblcf2emwj6ihkuOther Versions
Partial differential equations (PDEs) are central to describing and modelling complex physical systems that arise in many disciplines across science and engineering. ...
by partial differential equations. ...
Line 8 defines the functional F as a function of given control(s), which enables to only traverse the relevant part of F's computational graph needed to differentiate F with respect to the given control ...
arXiv:2303.06871v3
fatcat:wbwmvdt2x5bxlffolw3xbjq6oq
New directions in the applications of rough path theory
[article]
2023
arXiv
pre-print
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Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. ...
We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. ...
Section 2 introduces controlled differential equations (CDEs) and signatures. ...
arXiv:2302.04586v1
fatcat:wdw2betqd5bklmpfxy6g4752li
Neural Differential Equations as a Basis for Scientific Machine Learning (SciML)
[article]
2020
figshare.com
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Problems such as optimal control and automated learning of differential equation models will be reduced to training problems on neural differential equations. ...
In this talk we will introduce the audience to these methods and show how these diverse methods are all instantiations of a neural differential equation, a differential equation where all or part of the ...
16
Latent (Neural)
Differential Equations
NEURAL ORDINARY DIFFERENTIAL EQUATION:
′ = ( , , )
LET BE A NEURAL NETWORK
17
Training a neural differential equation
Solve the differential equation ...
doi:10.6084/m9.figshare.12751955.v1
fatcat:zhwjvt23tfhmjljetsfobsv5q4
Physics-Informed Tensor-Train ConvLSTM for Volumetric Velocity Forecasting of the Loop Current
2021
Frontiers in Artificial Intelligence
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Specifically, we propose (1) a novel 4D higher-order recurrent neural network with empirical orthogonal function analysis to capture the hidden uncorrelated patterns of each hierarchy, (2) a convolutional ...
tensor-train decomposition to capture higher-order space-time correlations, and (3) a mechanism that incorporates prior physics from domain experts by informing the learning in latent space. ...
The ranks here control the number of parameters in the tensor-train format. ...
doi:10.3389/frai.2021.780271
pmid:35005615
pmcid:PMC8741277
fatcat:ljz4h74skfbn7n3vslfpactszu
Quantum-Inspired Tensor Neural Networks for Partial Differential Equations
[article]
2022
arXiv
pre-print
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Partial Differential Equations (PDEs) are used to model a variety of dynamical systems in science and engineering. ...
To tackle these shortcomings, we implement Tensor Neural Networks (TNN), a quantum-inspired neural network architecture that leverages Tensor Network ideas to improve upon deep learning approaches. ...
Neural Networks for PDE The connection between PDE and Forward Backward Stochastic Differential Equations (FBSDE) is well studied in Refs. [6, 2, 18, 22, 8] . ...
arXiv:2208.02235v2
fatcat:tm7pqwfykzdbtahgh2fdwbs6iy
Noise-tolerant continuous-time Zhang neural networks for time-varying Sylvester tensor equations
2019
Advances in Difference Equations
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In this paper, to solve the time-varying Sylvester tensor equations (TVSTEs) with noise, we will design three noise-tolerant continuous-time Zhang neural networks (NTCTZNNs), termed NTCTZNN1, NTCTZNN2, ...
For comparison, the gradient-based neural network (GNN) is also presented and analyzed to solve TVSTEs. ...
In practice, various kinds of tensor equations arise from physics, mechanics, Markov process and partial differential equations. ...
doi:10.1186/s13662-019-2406-8
fatcat:6yzd72fbivav3hwbki5ldbjqle
Explainable Tensorized Neural Ordinary Differential Equations forArbitrary-step Time Series Prediction
[article]
2020
arXiv
pre-print
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We propose a continuous neural network architecture, termed Explainable Tensorized Neural Ordinary Differential Equations (ETN-ODE), for multi-step time series prediction at arbitrary time points. ...
Specifically, ETN-ODE combines an explainable Tensorized Gated Recurrent Unit (Tensorized GRU or TGRU) with Ordinary Differential Equations (ODE). ...
The integral could be obtained through a black-box differential equation solver. ...
arXiv:2011.13174v1
fatcat:4uxooploczestnbjlbvrl5xl7a
pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-dependent Partial Differential Equations
[article]
2024
arXiv
pre-print
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We present partial evolutionary tensor neural networks (pETNNs), a novel framework for solving time-dependent partial differential equations with both of high accuracy and remarkable extrapolation. ...
Our proposed architecture leverages the inherent accuracy of tensor neural networks, while incorporating evolutionary parameters that enable remarkable extrapolation capabilities. ...
Figure 1 : 1 Figure 1: Schematic diagram of the tensor neural networks (TNNs).
Figure 2 : 2 Figure 2: Schematic diagram of the partial evolutionary tensor neural networks (pETNNs). ...
arXiv:2403.06084v1
fatcat:c2rdulu5q5fo3mtbflhuufhwky
TT-PINN: A Tensor-Compressed Neural PDE Solver for Edge Computing
[article]
2022
arXiv
pre-print
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Physics-informed neural networks (PINNs) have been increasingly employed due to their capability of modeling complex physics systems. ...
To enable training PINNs on edge devices, this paper proposes an end-to-end compressed PINN based on Tensor-Train decomposition. ...
Introduction Physics-informed neural networks (PINNs) are increasingly used to solve a wide range of forward and inverse problems involving partial differential equations (PDEs), including fluids mechanics ...
arXiv:2207.01751v1
fatcat:kgkssa7j5zf4blittdacsegqp4
Neural Rough Differential Equations for Long Time Series
[article]
2021
arXiv
pre-print
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Neural controlled differential equations (CDEs) are the continuous-time analogue of recurrent neural networks, as Neural ODEs are to residual networks, and offer a memory-efficient continuous-time way ...
This is the approach for solving rough differential equations (RDEs), and correspondingly we describe our main contribution as the introduction of Neural RDEs. ...
Neural CDEs are similar to neural ordinary differential equations (ODEs), as popularised by . ...
arXiv:2009.08295v4
fatcat:3kc5uqy7krgabev75ms4go2tce
Solving high-dimensional parabolic PDEs using the tensor train format
[article]
2021
arXiv
pre-print
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In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and ...
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. ...
We would like to thank Reinhold Schneider for giving valuable input and for sharing his broad insight in tensor methods and optimization. ...
arXiv:2102.11830v2
fatcat:m4hrv45wn5cq3ppic2qpkttkii
Reduction and decomposition of differential automata: Theory and applications
[chapter]
1998
Lecture Notes in Computer Science
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John; Partial Differential Equations W. E. ...
Levinson; Theory of Ordinary Differential
Equation.
P. F. Hsieh and Y. Sibuya; Basic Theory of Ordinary Differential Equation.
M. K. Jain; Numerical Solution of Ordinary Differential Equation. ...
doi:10.1007/3-540-64358-3_48
fatcat:hqwvar3zbfftdhg4cybpjwogde
CPAC-Conv: CP-decomposition to Approximately Compress Convolutional Layers in Deep Learning
[article]
2020
arXiv
pre-print
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Feature extraction for tensor data serves as an important step in many tasks such as anomaly detection, process monitoring, image classification, and quality control. ...
Although the most recent methods in deep learning, such as Convolutional Neural Network (CNN), have shown outstanding performance in analyzing tensor data, their wide adoption is still hindered by model ...
At first, we calculate the differential of equation (10) at both sides w.r.t. ...
arXiv:2005.13746v1
fatcat:5hjmbfzpabdkpbp2zeilpiynue
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