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PDE+: Enhancing Generalization via PDE with Adaptive Distributional Diffusion
[article]
2023
arXiv
pre-print
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In the context of neural networks, we put this theoretical framework into practice as PDE+ (PDE with Adaptive Distributional Diffusion) which diffuses each sample into a distribution covering semantically ...
Building upon this, we propose a general framework that introduces adaptive distributional diffusion into transport equation to enhance the smoothness of its solution, thereby improving generalization. ...
An instantiation of this framework, called PDE+ can enhance the generalization via taking the augmented samples as semantic similar samples to guide the learning of adaptive distributional diffusion. ...
arXiv:2305.15835v2
fatcat:4atukchkz5fi7n4dssev6d6xee
Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
[article]
2024
arXiv
pre-print
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2023 pre-print arXiv:2302.05104v1 fatcat:6e4kfin7nbcszexcad7ofoibbyOther Versions
These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. ...
In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. ...
C.2.3 Heat Diffusion on a Circular Ring The ground-truth data is generated via the Python package 'py-pde' [62] , and the step size is fixed as 10 −4 . ...
arXiv:2302.05104v4
fatcat:epk3qqrkqfhrbeyp4ivx4vxw7i
PDEs for Tensor Image Processing
[chapter]
2006
Mathematics and Visualization
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We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. ...
Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. ...
Acknowledgements We are grateful to Anna Vilanova i Bartrolí (Eindhoven Institute of Technology) and Carola van Pul (Maxima Medical Center, Eindhoven) for providing us with the DT-MRI dataset and for discussing ...
doi:10.1007/3-540-31272-2_25
fatcat:z5bwkbtfafdv3nwhtvszzgfnwa
PDE-Driven Adaptive Morphology for Matrix Fields
[chapter]
2009
Lecture Notes in Computer Science
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In this article we propose a novel method to incorporate adaptivity into the matrix-valued, PDE-driven dilation process. ...
A prominent example is diffusion tensor magnetic resonance imaging (DT-MRI) which is a medical imaging technique useful for analysing the fibre structure in the brain. ...
In all our experiments we compare the results of the proposed matrix-valued adaptive dilation with the isotropic dilation [8] , and with the matrix-valued coherence-enhancing diffusion from [10] . ...
doi:10.1007/978-3-642-02256-2_21
fatcat:3h6iql2gvndi3ass7pftslfogq
Selective Extraction of Entangled Textures via Adaptive PDE Transform
2012
International Journal of Biomedical Imaging
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The proposed adaptive PDE transform is applied to the selective extraction of entangled textures. ...
The present work introduces adaptive PDE transform algorithm to appropriately threshold the statistical variance of the local variation of functional modes. ...
diffusion coefficients and enhancement operator respectively. ...
doi:10.1155/2012/958142
pmid:22315584
pmcid:PMC3272340
fatcat:vq7uxin23bfhdctpyixpenks6a
New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes
2009
Measurement science and technology
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The goal of such a new adaptive diffusion scheme is to better preserve edges when the anisotropic diffusion PDE models are applied to image enhancement tasks. ...
The proposed adaptive parameter estimation in the anisotropic diffusion PDE model involves self-organizing maps and Bayesian inference to define edge probabilities accurately. ...
The goal of such a new adaptive diffusion scheme is to better preserve edges when the anisotropic diffusion PDE models are applied to image enhancement. ...
doi:10.1088/0957-0233/20/10/104012
fatcat:wylbbdb2ybd57c6pnaihc3b4r4
Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems
[article]
2021
arXiv
pre-print
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Furthermore, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the ...
However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. ...
gPINN enhanced by RAR To further improve the accuracy and training efficiency of gPINN for solving PDEs with a stiff solution, we apply RAR to adaptively improve the distribution of residual points during ...
arXiv:2111.02801v1
fatcat:eeehsuumlvgedfjpdakuua2mne
Accelerating PDE Data Generation via Differential Operator Action in Solution Space
[article]
2024
arXiv
pre-print
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2024 pre-print arXiv:2402.05957v1 fatcat:bqzuwxnsgvahna43dg3azdxjxq
To address this challenge, we propose a novel PDE dataset generation algorithm, namely Differential Operator Action in Solution space (DiffOAS), which speeds up the data generation process and enhances ...
Experimental results show that DiffOAS accelerates the generation of large-scale datasets with 10,000 instances by 300 times. ...
q(x, y) is generated by uniform distribution U [0, 1]. ...
arXiv:2402.05957v2
fatcat:x74qq4xqpnd3xladzbbedukob4
Solving inverse-PDE problems with physics-aware neural networks
[article]
2020
arXiv
pre-print
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We call this architecture Blended inverse-PDE networks (hereby dubbed BiPDE networks) and demonstrate its applicability for recovering the variable diffusion coefficient in Poisson problems in one and ...
We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom ...
We consider random diffusion coefficients a 0 and a 1 with a uniform distribution in [0.25, 0.75] and we generate 1000 solutions over the domain x ∈ [−1, 1]. ...
arXiv:2001.03608v3
fatcat:hkhelkijvzawvbsfdoyj52wwfu
SPRINT2D Software for Convection Dominated PDEs
[chapter]
1997
Modern Software Tools for Scientific Computing
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The software uses unstructured triangular meshes and adaptive error control in both space and time. ...
This chapter describes the software and shows how the adaptive techniques may be used to increase the reliability of the solution procedure for a challenging combustion problem. ...
This information is then transferred to mesh generation software via a postprocessing routine. ...
doi:10.1007/978-1-4612-1986-6_3
dblp:conf/scitools/BerzinsPPW96
fatcat:thgyvsymtfd5zipzjfwesfz2iy
PDE-Driven Spatiotemporal Disentanglement
[article]
2021
arXiv
pre-print
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Following this direction, we propose in this article a novel and general paradigm for this task based on a resolution method for partial differential equations: the separation of variables. ...
Patrick Gallinari is additionally funded by the 2019 ANR AI Chairs program via the DL4CLIM project. ...
It helps to solve refinements of the heat equation, such as generalizations with an advection term (see Appendix A.2) or with complex diffusion and source terms forming a general transport equation (Jia ...
arXiv:2008.01352v3
fatcat:di3nuh7jcnggdm2px44hiukb4m
Anisotropic Elliptic PDEs for Feature Classification
2013
IEEE Transactions on Visualization and Computer Graphics
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At the computational front, the heat diffusion PDE becomes a linear system with Dirichlet condition at heat sources (called seeds). ...
We tackle this challenge by exploring a quasiharmonic field (QHF) generated by elliptic PDEs, which is the stable state of heat diffusion governed by anisotropic diffusion tensor. ...
In addition, the steady state of the heat diffusion corresponds to a quasi-harmonic field (QHF) generated by an elliptic PDE with the weighted diffusion operators. ...
doi:10.1109/tvcg.2013.60
pmid:23929843
fatcat:pk7gstbhwvbl7nx24ud26kvpee
Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution
2015
PLoS ONE
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In the first method we use the enhancement PDE for contextual regularization of a fiber orientation distribution (FOD) that is obtained on individual voxels from high angular resolution diffusion imaging ...
Both methods are based on the same PDE framework, defined in the coupled space of positions and orientations, associated with a stochastic process describing the enhancement of elongated structures while ...
Roebroeck from the Faculty of Psychology & Neuroscience, Maastricht University for providing us with the human data used in Section 3.2. ...
doi:10.1371/journal.pone.0138122
pmid:26465600
pmcid:PMC4605742
fatcat:rhvdzlqvnrea3lvcijpyf7yxta
Self-Adaptive Methods for PDE
2004
Oberwolfach Reports
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With the incorporation of more and more complete physics these models become increasingly complex and their accurate numerical simulation requires the use of efficient self-adaptive methods. ...
of adaptive finite element methods. • Multiscale adaptivity and wavelets. • Space-time adaptivity in nonstationary problems. ...
We ask if the accuracy of the discretization can be enhanced by adapting the diffusion parameters locally. ...
doi:10.4171/owr/2004/16
fatcat:5ywgyr4v3fho7gcmrfdrgfjcnu
Combining Image Space and q-Space PDEs for Lossless Compression of Diffusion MR Images
[article]
2022
arXiv
pre-print
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Diffusion MRI is a modern neuroimaging modality with a unique ability to acquire microstructural information by measuring water self-diffusion at the voxel level. ...
However, it generates huge amounts of data, resulting from a large number of repeated 3D scans. ...
The general idea behind this approach is to store a sparse subset of the image information, and to reconstruct the remaining image via PDE-based inpainting [2] . ...
arXiv:2206.06846v1
fatcat:dparcw4iu5fghc56k7ypuvwtu4
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