Feature Engineering with Regularity Structures

Chevyrev, Ilya; Gerasimovics, Andris; Weber, H.

doi:10.48550/arxiv.2108.05879

Cited by 2 publications

(12 citation statements)

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“…In the rough path theory, path signature can well characterize the path up to a natural equivalence relation. Model feature vectors (Chevyrev, Gerasimovics, and Weber 2021) are the multi-dimensional generalization of path signature, i.e., from the temporal dimension to the spatial-temporal dimension. Both path signature and model feature vectors are used as features in applications for modelling spatial-temporal data, such as a solution of CDE (Morrill et al 2021) and SPDE (Chevyrev, Gerasimovics, and Weber 2021;Hu et al 2022) respectively.…”

Section: Related Workmentioning

confidence: 99%

“…The model feature vectors (Chevyrev, Gerasimovics, and Weber 2021) inspired by the regularity structure theory (Hairer 2014b) compose a set of bases of SPDEs' mild solution. They have improved regularity compared to the random forcing and have previously been treated as a fixed…”

Section: Introductionmentioning

confidence: 99%

“…The Thirty-Seventh AAAI Conference on Artificial Intelligence feature transformation, on top of which a model may be built. (Chevyrev, Gerasimovics, and Weber 2021) was the first work to propose the model feature vector for learning the SPDEs. However, (Chevyrev, Gerasimovics, and Weber 2021) used the regularity structure as a deterministic feature extraction and only considered the linear model, which significantly limits the effectiveness of this feature to predict the SPDE solution.…”

Section: Introductionmentioning

confidence: 99%

“…(Chevyrev, Gerasimovics, and Weber 2021) was the first work to propose the model feature vector for learning the SPDEs. However, (Chevyrev, Gerasimovics, and Weber 2021) used the regularity structure as a deterministic feature extraction and only considered the linear model, which significantly limits the effectiveness of this feature to predict the SPDE solution. Prior knowledge is needed to determine the number of feature vectors, and its computational cost grows exponentially as the order of the features becomes large.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Gong

Wang

et al. 2023

AAAI

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Stochastic partial differential equations (SPDEs) are crucial for modelling dynamics with randomness in many areas including economics, physics, and atmospheric sciences. Recently, using deep learning approaches to learn the PDE solution for accelerating PDE simulation becomes increasingly popular. However, SPDEs have two unique properties that require new design on the models. First, the model to approximate the solution of SPDE should be generalizable over both initial conditions and the random sampled forcing term. Second, the random forcing terms usually have poor regularity whose statistics may diverge (e.g., the space-time white noise). To deal with the problems, in this work, we design a deep neural network called \emph{Deep Latent Regularity Net} (DLR-Net). DLR-Net includes a regularity feature block as the main component, which maps the initial condition and the random forcing term to a set of regularity features. The processing of regularity features is inspired by regularity structure theory and the features provably compose a set of basis to represent the SPDE solution. The regularity features are then fed into a small backbone neural operator to get the output. We conduct experiments on various SPDEs including the dynamic $\Phi^4_1$ model and the stochastic 2D Navier-Stokes equation to predict their solutions, and the results demonstrate that the proposed DLR-Net can achieve SOTA accuracy compared with the baselines. Moreover, the inference time is over 20 times faster than the traditional numerical solver and is comparable with the baseline deep learning models.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Gong

Wang

et al. 2023

AAAI

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“…The need for nonlinear functionals of the field x in texture classification was acknowledged in the influential paper [SM14], leading to a wide variety of generalizations. One can also mention the recent contribution [CGW21], where regularity structures based features are used for prediction purposes. However, the fact that 2-d signatures are natural objects to consider for image processing is not mentioned in those two references.…”

Section: Introductionmentioning

confidence: 99%

2-d signature of images and texture classification

Zhang¹,

Lin²,

Tindel³

2022

Preprint

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We introduce a proper notion of 2-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a 2-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here we implement a simple procedure for texture classification. In this context, we show that a low dimensional set of features based on signatures produces an excellent accuracy.

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Feature Engineering with Regularity Structures

Cited by 2 publications

References 0 publications

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

2-d signature of images and texture classification

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