Chi Chiu So

Chi Chiu So, Siu Pang Yung

Finding solutions to partial differential equations (PDEs) is an important and essential component in many scientific and engineering discoveries. One of the common approaches empowered by deep learning is Physics-informed Neural Networks (PINNs). Recently, a new type of fundamental neural network model, Kolmogorov-Arnold Networks (KANs), has been proposed as a substitute of Multilayer Perceptions (MLPs), and possesses trainable activation functions. To enhance KANs in fitting accuracy, a modification of KANs, so called ReLU-KANs, using "square of ReLU" as the basis of its activation functions, has been suggested. In this work, we propose another basis of activation functions, namely, Higherorder-ReLU (HR), which is simpler than the basis of activation functions used in KANs, namely, Bsplines; allows efficient KAN matrix operations; and possesses smooth and non-zero higher-order derivatives, essential to physicsinformed neural networks. We name such KANs with Higher-order-ReLU (HR) as their activations, HRKANs. Our detailed experiments on two famous and representative PDEs, namely, the linear Poisson equation and nonlinear Burgers' equation with viscosity, reveal that our proposed Higher-order-ReLU-KANs (HRKANs) achieve the highest fitting accuracy and training robustness and lowest training time significantly among KANs, ReLU-KANs and HRKANs. The codes to replicate our experiments are available at https://github.com/kelvinhkcs/HRKAN.

Yuan-dong Cao, Chi Chiu SO, Jun-Min Wang et al.

Physics-informed neural networks (PINNs) are powerful surrogates for differential equations but are notoriously difficult to train due to spectral bias, stiffness, and poor accuracy on high-frequency or multiscale solutions. Adversarial training based on generative adversarial networks (GANs) has recently gained surprisingly strong empirical results in improving training, but the underlying mechanisms remain elusive. To this end, we propose a new analysis framework for adversarially trained PINNs, based on the key observation of how the discriminator in GANs can influence the training dynamics of PINNs. The framework first provides a much needed theoretical grounding to why and when adversarial training is effective in PINNs, then presents a unified analysis of GANs variants in such training, and finally leads to a new, practical, efficient training algorithm for PINNs. Empirical results demonstrate that our method can significantly reduce the pathology of PINNs training, thereby providing better models with superior performances, often several magnitudes more accurate than alternative methods.

Chi Chiu So, Yueyue Sun, Jun-Min Wang et al.

How far are Large Language Models (LLMs) in performing deep relational reasoning? In this paper, we evaluate and compare the reasoning capabilities of three cutting-edge LLMs, namely, DeepSeek-R1, DeepSeek-V3 and GPT-4o, through a suite of carefully designed benchmark tasks in family tree and general graph reasoning. Our experiments reveal that DeepSeek-R1 consistently achieves the highest F1-scores across multiple tasks and problem sizes, demonstrating strong aptitude in logical deduction and relational inference. However, all evaluated models, including DeepSeek-R1, struggle significantly as problem complexity increases, largely due to token length limitations and incomplete output structures. A detailed analysis of DeepSeek-R1's long Chain-of-Thought responses uncovers its unique planning and verification strategies, but also highlights instances of incoherent or incomplete reasoning, calling attention to the need for deeper scrutiny into LLMs' internal inference dynamics. We further discuss key directions for future work, including the role of multimodal reasoning and the systematic examination of reasoning failures. Our findings provide both empirical insights and theoretical implications for advancing LLMs' reasoning abilities, particularly in tasks that demand structured, multi-step logical inference. Our code repository will be publicly available at https://github.com/kelvinhkcs/Deep-Relational-Reasoning.