用神经泰勒展开实现时间自适应的偏微分方程求解，误差更小、速度更快。
TANTE: Time-Adaptive Operator Learning via Neural Taylor Expansion
- 通过神经网络学习高阶时间导数和收敛半径，动态调整预测步长。
- 在多个PDE基准上实现60-80%精度提升，推理速度加快30-40%。
- 适合需要高精度与高效计算的物理模拟场景，如气候建模、流体仿真。
针对时变偏微分方程（PDE）的算子学习近年来发展迅速，可高效逼近复杂的时空动力学。然而，现有方法多采用固定时间步长进行滚动预测，难以适应变化的时间复杂度，常导致误差累积。本文提出时间自适应变换器与神经泰勒展开结合的TANTE框架，实现连续时间预测并动态调整步长。TANTE在当前状态执行泰勒展开，由神经网络学习高阶时间导数及局部收敛半径，使模型能根据解的局部行为自适应调整滚动策略，从而减少累积误差并提升计算效率。我们在多个典型PDE基准上验证了TANTE的有效性，相比固定步长基线，在推理阶段实现60-80%的精度提升和30-40%的速度加速。代码已开源：https://github.com/zwu88/TANTE，确保透明与可复现。
原文摘要 · Abstract (English)
Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step sizes during rollout, which limits their ability to adapt to varying temporal complexity and often leads to error accumulation. Here, we propose the Time-Adaptive Transformer with Neural Taylor Expansion (TANTE), a novel operator-learning framework that produces continuous-time predictions with adaptive step sizes. TANTE predicts future states by performing a Taylor expansion at the current state, where neural networks learn both the higher-order temporal derivatives and the local radius of convergence. This allows the model to dynamically adjust its rollout based on the local behavior of the solution, thereby reducing cumulative error and improving computational efficiency. We demonstrate the effectiveness of TANTE across a wide range of PDE benchmarks, achieving superior accuracy and adaptability compared to fixed-step baselines, delivering accuracy gains of 60-80 % and speed-ups of 30-40 % at inference time. The code is publicly available at https://github.com/zwu88/TANTE for transparency and reproducibility.
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