From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8\% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

Controllable Textual Inversion for Personalized Text-to-Image Generation

The recent large-scale generative modeling has attained unprecedented performance especially in producing high-fidelity images driven by text prompts. Text inversion (TI), alongside the text-to-image model backbones, is proposed as an effective technique in personalizing the generation when the prompts contain user-defined, unseen or long-tail concept tokens. Despite that, we find and show that the deployment of TI remains full of "dark-magics" -- to name a few, the harsh requirement of additional datasets, arduous human efforts in the loop and lack of robustness. In this work, we propose a much-enhanced version of TI, dubbed Controllable Textual Inversion (COTI), in resolving all the aforementioned problems and in turn delivering a robust, data-efficient and easy-to-use framework. The core to COTI is a theoretically-guided loss objective instantiated with a comprehensive and novel weighted scoring mechanism, encapsulated by an active-learning paradigm. The extensive results show that COTI significantly outperforms the prior TI-related approaches with a 26.05 decrease in the FID score and a 23.00% boost in the R-precision.

A critical look at the current train/test split in machine learning

The randomized or cross-validated split of training and testing sets has been adopted as the gold standard of machine learning for decades. The establishment of these split protocols are based on two assumptions: (i)-fixing the dataset to be eternally static so we could evaluate different machine learning algorithms or models; (ii)-there is a complete set of annotated data available to researchers or industrial practitioners. However, in this article, we intend to take a closer and critical look at the split protocol itself and point out its weakness and limitation, especially for industrial applications. In many real-world problems, we must acknowledge that there are numerous situations where assumption (ii) does not hold. For instance, for interdisciplinary applications like drug discovery, it often requires real lab experiments to annotate data which poses huge costs in both time and financial considerations. In other words, it can be very difficult or even impossible to satisfy assumption (ii). In this article, we intend to access this problem and reiterate the paradigm of active learning, and investigate its potential on solving problems under unconventional train/test split protocols. We further propose a new adaptive active learning architecture (AAL) which involves an adaptation policy, in comparison with the traditional active learning that only unidirectionally adds data points to the training pool. We primarily justify our points by extensively investigating an interdisciplinary drug-protein binding problem. We additionally evaluate AAL on more conventional machine learning benchmarking datasets like CIFAR-10 to demonstrate the generalizability and efficacy of the new framework.