CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations
This work addresses the challenge of efficient and accurate modeling in data-scarce regimes for physics and engineering applications, representing an incremental improvement over existing neural network methods.
The paper tackles the problem of predicting solution fields for parameterized partial differential equations by introducing CoLoRA, a method that pre-trains neural networks and continuously adapts low-rank weights in time, achieving orders-of-magnitude faster predictions and higher accuracy and parameter efficiency compared to other neural network approaches.
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.