This work introduces reduced models based on Continuous Low Rank Adaptation (CoLoRA) that pre-train neural networks for a given partial differential equation and then continuously adapt low-rank weights in time to rapidly predict the evolution of solution fields at new physics parameters and new initial conditions. The adaptation can be either purely data-driven or via an equation-driven variational approach that provides Galerkin-optimal approximations. Because CoLoRA approximates solution fields locally in time, the rank of the weights can be kept small, which means that only few training trajectories are required offline so that CoLoRA is well suited for data-scarce regimes. Predictions with CoLoRA are orders of magnitude faster than with classical methods and their accuracy and parameter efficiency is higher compared to other neural network approaches.

翻译：本文提出基于连续低秩自适应（CoLoRA）的约简模型，该模型针对给定偏微分方程预训练神经网络，并通过时间维度的连续低秩权重自适应，快速预测新物理参数及初值条件下解场的演化过程。自适应策略既可采用纯数据驱动方式，也可通过基于方程的变分方法实现Galerkin最优逼近。由于CoLoRA在时间维度上对解场进行局部逼近，权重秩可保持较小规模，这意味着离线阶段仅需少量训练轨迹即可完成重构，因此特别适用于数据稀缺场景。与经典方法相比，CoLoRA的预测速度提升数个数量级，且其精度与参数效率均优于其他神经网络方法。