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Transformer for Partial Differential Equations' Operator Learning
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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.
Forward citations
Cited by 15 Pith papers
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Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators
A single Fourier Neural Operator can learn the full state dynamics of stiff ionic models up to 41 variables with roughly 2% relative L2 test error.
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Sequential Neural Operator Transformer for High-Fidelity Surrogates of Time-Dependent Non-linear Partial Differential Equations
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GITO: Graph-Informed Transformer Operator for Learning Complex Partial Differential Equations
GITO, a graph-informed transformer operator, reports lower relative L2 errors than existing transformer-based neural operators on Navier-Stokes, heat conduction, and airfoil benchmark datasets.
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Reference neural operators with a Virtual-Fourier layer learn solution derivatives and a hybrid solver feedback loop accelerates PDE-constrained optimization.
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Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery
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Efficient Transformer-Inspired Variants of Physics-Informed Deep Operator Networks
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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Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.
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