This work presents a novel neural network architecture called the "volume-preserving transformer" that is designed to learn the dynamics of systems described by divergence-free vector fields. The key innovations are:
The standard transformer attention mechanism is replaced with a volume-preserving attention layer that preserves the volume of the input space. This is achieved by using the Cayley transform to ensure the attention weights form an orthogonal matrix.
The feedforward neural network component of the transformer is replaced with a volume-preserving feedforward network, which uses lower and upper triangular weight matrices to guarantee volume preservation.
The authors demonstrate the effectiveness of the volume-preserving transformer on the example of the rigid body dynamics, which is described by a divergence-free vector field. Compared to a standard transformer and a volume-preserving feedforward network, the volume-preserving transformer shows superior performance in accurately capturing the long-term dynamics of the system.
The authors also discuss the importance of incorporating physical properties, such as volume preservation, into neural network architectures for modeling dynamical systems. They highlight that this is crucial for ensuring stable and physically meaningful predictions, especially when applying the models to real-world applications.
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by Benedikt Bra... kl. arxiv.org 05-02-2024
https://arxiv.org/pdf/2312.11166.pdfDybere Forespørgsler