Junping Yin

Xinxin Li, Xingyu Cui, Jin Qi et al.

Discovering governing Partial Differential Equations (PDEs) from sparse and noisy data is a challenging issue in data-driven scientific computing. Conventional sparse regression methods often suffer from two major limitations: (i) the instability of numerical differentiation under sparse and noisy data, and (ii) the restricted flexibility of a pre-defined candidate library. We propose Weak-PDE-Net, an end-to-end differentiable framework that can robustly identify open-form PDEs. Weak-PDE-Net consists of two interconnected modules: a forward response learner and a weak-form PDE generator. The learner embeds learnable Gaussian kernels within a lightweight MLP, serving as a surrogate model that adaptively captures system dynamics from sparse observations. Meanwhile, the generator integrates a symbolic network with an integral module to construct weak-form PDEs, avoiding explicit numerical differentiation and improving robustness to noise. To relax the constraints of the pre-defined library, we leverage Differentiable Neural Architecture Search strategy during training to explore the functional space, which enables the efficient discovery of open-form PDEs. The capability of Weak-PDE-Net in multivariable systems discovery is further enhanced by incorporating Galilean Invariance constraints and symmetry equivariance hypotheses to ensure physical consistency. Experiments on several challenging PDE benchmarks demonstrate that Weak-PDE-Net accurately recovers governing equations, even under highly sparse and noisy observations.

Da Li, Junping Yin, Jin Xu et al.

Extracting simple mathematical expression from an observational dataset to describe complex natural phenomena is one of the core objectives of artificial intelligence (AI). This field is known as symbolic regression (SR). Traditional SR models are based on genetic programming (GP) or reinforcement learning (RL), facing well-known challenges, such as low efficiency and overfitting. Recent studies have integrated SR with large language models (LLMs), enabling fast zero-shot inference by learning mappings from millions of dataset-expression pairs. However, since the input and output are inherently different modalities, such models often struggle to converge effectively. In this paper, we introduce ViSymRe, a vision-guided multimodal SR model that incorporates the third resource, expression graph, to bridge the modality gap. Different from traditional multimodal models, ViSymRe is trained to extract vision, termed virtual vision, from datasets, without relying on the global availability of expression graphs, which addresses the essential challenge of visual SR, i.e., expression graphs are not available during inference. Evaluation results on multiple mainstream benchmarks show that ViSymRe achieves more competitive performance than the state-of-the-art dataset-only baselines. The expressions predicted by ViSymRe not only fit the dataset well but are also simple and structurally accurate, goals that SR models strive to achieve.

Xinxin Li, Juan Zhang, Da Li et al.

Symbolic Regression (SR) is a powerful technique for automatically discovering mathematical expressions from input data. Mainstream SR algorithms search for the optimal symbolic tree in a vast function space, but the increasing complexity of the tree structure limits their performance. Inspired by neural networks, symbolic networks have emerged as a promising new paradigm. However, most existing symbolic networks still face certain challenges: binary nonlinear operators $\{\times, ÷\}$ cannot be naturally extended to multivariate operators, and training with fixed architecture often leads to higher complexity and overfitting. In this work, we propose a Unified Symbolic Network that unifies nonlinear binary operators into nested unary operators and define the conditions under which UniSymNet can reduce complexity. Moreover, we pre-train a Transformer model with a novel label encoding method to guide structural selection, and adopt objective-specific optimization strategies to learn the parameters of the symbolic network. UniSymNet shows high fitting accuracy, excellent symbolic solution rate, and relatively low expression complexity, achieving competitive performance on low-dimensional Standard Benchmarks and high-dimensional SRBench.