Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems

Predictor feedback designs are critical for delay-compensating controllers in nonlinear systems. However, these designs are limited in practical applications as predictors cannot be directly implemented, but require numerical approximation schemes. These numerical schemes, typically combining finite difference and successive approximations, become computationally prohibitive when the dynamics of the system are expensive to compute. To alleviate this issue, we propose approximating the predictor mapping via a neural operator. In particular, we introduce a new perspective on predictor designs by recasting the predictor formulation as an operator learning problem. We then prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated-predictor, we achieve semiglobal practical stability of the closed-loop nonlinear system. The estimate is semiglobal in a unique sense - namely, one can increase the set of initial states as large as desired but this will naturally increase the difficulty of training a neural operator approximation which appears practically in the stability estimate. Furthermore, we emphasize that our result holds not just for neural operators, but any black-box predictor satisfying a universal approximation error bound. From a computational perspective, the advantage of the neural operator approach is clear as it requires training once, offline and then is deployed with very little computational cost in the feedback controller. We conduct experiments controlling a 5-link robotic manipulator with different state-of-the-art neural operator architectures demonstrating speedups on the magnitude of $10^2$ compared to traditional predictor approximation schemes.

BiDoRA: Bi-level Optimization-Based Weight-Decomposed Low-Rank Adaptation

Parameter-efficient fine-tuning (PEFT) of large language models (LLMs) has gained considerable attention as a flexible and efficient way of adapting LLMs to downstream tasks. Among these methods, weighted decomposed low-rank adaptation (DoRA) has emerged as a promising approach. DoRA bridges the gap between low-rank adaptation (LoRA) and full fine-tuning (FT) by decomposing the weight matrices into magnitude and direction components, thereby maintaining learning behavior similar to FT. Although DoRA shows encouraging performance, it introduces additional parameters compared to LoRA, which potentially increases the risk of overfitting. Moreover, optimizing magnitude and direction simultaneously leads to a coupled gradient updating pattern for both components, limiting its learning capacity. To overcome these limitations, we propose BiDoRA, a bi-level optimization-based PEFT method. In BiDoRA, the direction and magnitude components are optimized on two distinct datasets at different optimization levels, mitigating the risk of overfitting. Additionally, the asynchronous optimization of the two components promotes their decoupling, allowing for more flexible gradient updates suitable for various downstream tasks. Evaluation of BiDoRA on fourteen datasets spanning natural language understanding, natural language generation, and token classification reveals that it significantly outperforms DoRA and other PEFT methods. The superior performance of BiDoRA underscores its effectiveness. The code for BiDoRA is available at https://anonymous.4open.science/r/BiDoRA-5D31.