Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Meta-Auto-Decoder for Solving Parametric Partial Differential Equations   Partial Differential Equations (PDEs) are ubiquitous in many disciplines of
science and engineering and notoriously difficult to solve. In general,
closed-form solutions of PDEs are unavailable and numerical approximation
methods are computationally expensive. The parameters of PDEs are variable in
many applications, such as inverse problems, control and optimization, risk
assessment, and uncertainty quantification. In these applications, our goal is
to solve parametric PDEs rather than one instance of them. Our proposed
approach, called Meta-Auto-Decoder (MAD), treats solving parametric PDEs as a
meta-learning problem and utilizes the Auto-Decoder structure in
\cite{park2019deepsdf} to deal with different tasks/PDEs. Physics-informed
losses induced from the PDE governing equations and boundary conditions is used
as the training losses for different tasks. The goal of MAD is to learn a good
model initialization that can generalize across different tasks, and eventually
enables the unseen task to be learned faster. The inspiration of MAD comes from
(conjectured) low-dimensional structure of parametric PDE solutions and we
explain our approach from the perspective of manifold learning. Finally, we
demonstrate the power of MAD though extensive numerical studies, including
Burgers' equation, Laplace's equation and time-domain Maxwell's equations. MAD
exhibits faster convergence speed without losing the accuracy compared with
other deep learning methods.

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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: math
23
The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci