Deep learning-based modeling of dynamical systems driven by partial differential equations (PDEs) has become quite popular in recent years. However, most of the existing deep learning-based methods either assume strong physics prior, or depend on specific initial and boundary conditions, or require data in dense regular grid making them inapt for modeling unknown PDEs from sparsely-observed data. This paper presents a deep learning-based collocation method for modeling dynamical systems driven by unknown PDEs when data sites are sparsely distributed . The proposed method is spatial dimension-independent , geometrically flexible, learns from sparsely-available data and the learned model does not depend on any specific initial and boundary conditions . We demonstrate our method in the forecasting task for  two-dimensional wave equation and  Burgers-Fisher equation in multiple geometries with different boundary conditions  .
In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs) . We propose a deep learning framework that learns the underlying dynamics and predicts its evolution using sparsely distributed data sites. Deep learning has shown promising results in modeling physical dynamics in recent years. However, most of the existing deep learning methods for modeling physical dynamics either focus on solving known PDEs or require data in a dense grid when the governing PDEs are unknown. In contrast, our method focuses on learning prediction models for unknown PDE-driven dynamics only from sparsely observed data . The proposed method is spatial dimension-independent and geometrically flexible . We demonstrate our method in the forecasting task for the two-dimensional wave equation and the Burgers-Fisher equation in multiple geometries with different boundary conditions , and the ten-dimensional heat equation .
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Deep learning-based modeling of dynamical systems driven by
In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown
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has become quite popular in
. We propose a deep learning framework that learns the underlying dynamics and predicts its evolution using sparsely distributed data sites. Deep learning has shown promising results in modeling physical dynamics in
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learning-based methods either assume strong physics prior, or depend on specific initial and boundary conditions,
learning methods for modeling physical dynamics either focus on solving known PDEs
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dense regular grid making them inapt for modeling unknown PDEs from sparsely-observed data. This paper presents a deep learning-based collocation method for modeling dynamical systems driven by unknown PDEs when data sites are sparsely distributed
a dense grid when the governing PDEs are unknown. In contrast, our method focuses on learning prediction models for unknown PDE-driven dynamics only from sparsely observed data
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, geometrically flexible, learns from sparsely-available data and the learned model does not depend on any specific initial and boundary conditions
and geometrically flexible
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, and the ten-dimensional heat equation