用神经算子建模时空控制系统，实现高效迁移与连续时间表示。
RHYME-XT: A Neural Operator for Spatiotemporal Control Systems
- 通过神经网络学习空间基函数，将复杂方程投影到低维空间。
- 直接学习系统流映射，避免积分计算，保持连续时间特性。
- 跨数据集微调后知识迁移效果好，性能优于现有最优模型。
我们提出RHYME-XT，一种用于输入仿射非线性偏积分微分方程（PIDE）驱动的具有局部节律行为的时空控制系统的代理建模神经算子框架。RHYME-XT采用伽辽金投影，利用由神经网络参数化的空间基函数对无限维PIDE进行有限维近似，得到由投影输入驱动的常微分方程组。不同于对非自治系统进行积分，我们直接使用学习流函数的架构来学习其流映射，避免了昂贵的计算，同时获得连续时间且离散化无关的表示。在神经场PIDE上的实验表明，RHYME-XT优于当前最先进的神经算子，并能通过微调过程在不同数据集训练的模型间有效传递知识。
原文摘要 · Abstract (English)
We propose RHYME-XT, an operator-learning framework for surrogate modeling of spatiotemporal control systems governed by input-affine nonlinear partial integro-differential equations (PIDEs) with localized rhythmic behavior. RHYME-XT uses a Galerkin projection to approximate the infinite-dimensional PIDE on a learned finite-dimensional subspace with spatial basis functions parameterized by a neural network. This yields a projected system of ODEs driven by projected inputs. Instead of integrating this non-autonomous system, we directly learn its flow map using an architecture for learning flow functions, avoiding costly computations while obtaining a continuous-time and discretization-invariant representation. Experiments on a neural field PIDE show that RHYME-XT outperforms a state-of-the-art neural operator and is able to transfer knowledge effectively across models trained on different datasets, through a fine-tuning process.
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