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Transformer for Partial Differential Equations' Operator Learning

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arxiv 2205.13671 v3 pith:G3FJRAXG submitted 2022-05-26 cs.LG cs.AI

Zijie Li , Kazem Meidani , Amir Barati Farimani This is my paper
classification cs.LGcs.AI
keywords learningframeworkoperatorbuiltdata-drivendifferentialequationsinput
verification ladder T0 review T1 audit T2 compute T3 formal
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Data-driven learning of partial differential equations' solution operators has recently emerged as a promising paradigm for approximating the underlying solutions. The solution operators are usually parameterized by deep learning models that are built upon problem-specific inductive biases. An example is a convolutional or a graph neural network that exploits the local grid structure where functions' values are sampled. The attention mechanism, on the other hand, provides a flexible way to implicitly exploit the patterns within inputs, and furthermore, relationship between arbitrary query locations and inputs. In this work, we present an attention-based framework for data-driven operator learning, which we term Operator Transformer (OFormer). Our framework is built upon self-attention, cross-attention, and a set of point-wise multilayer perceptrons (MLPs), and thus it makes few assumptions on the sampling pattern of the input function or query locations. We show that the proposed framework is competitive on standard benchmark problems and can flexibly be adapted to randomly sampled input.

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Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 46 citations worldwide. Full citation record

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    Six input-conditioned DeepONet variants match or approach modified DeepONet accuracy on four PDE benchmarks with roughly half the training time.

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