SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving

We propose SymPlex, a reinforcement learning framework for discovering analytical symbolic solutions to partial differential equations (PDEs) without access to ground-truth expressions. SymPlex formulates symbolic PDE solving as tree-structured decision-making and optimizes candidate solutions using only the PDE and its boundary conditions. At its core is SymFormer, a structure-aware Transformer that models hierarchical symbolic dependencies via tree-relative self-attention and enforces syntactic validity through grammar-constrained autoregressive decoding, overcoming the limited expressivity of sequence-based generators. Unlike numerical and neural approaches that approximate solutions in discretized or implicit function spaces, SymPlex operates directly in symbolic expression space, enabling interpretable and human-readable solutions that naturally represent non-smooth behavior and explicit parametric dependence. Empirical results demonstrate exact recovery of non-smooth and parametric PDE solutions using deep learning-based symbolic methods.

Uncertainty-Aware Flow Field Reconstruction Using SVGP Kolmogorov-Arnold Networks

Reconstructing time-resolved flow fields from temporally sparse velocimetry measurements is critical for characterizing many complex thermal-fluid systems. We introduce a machine learning framework for uncertainty-aware flow reconstruction using sparse variational Gaussian processes in the Kolmogorov-Arnold network topology (SVGP-KAN). This approach extends the classical foundations of Linear Stochastic Estimation (LSE) and Spectral Analysis Modal Methods (SAMM) while enabling principled epistemic uncertainty quantification. We perform a systematic comparison of our framework with the classical reconstruction methods as well as Kalman filtering. Using synthetic data from pulsed impingement jet flows, we assess performance across fractional PIV sampling rates ranging from 0.5% to 10%. Evaluation metrics include reconstruction error, generalization gap, structure preservation, and uncertainty calibration. Our SVGP-KAN methods achieve reconstruction accuracy comparable to established methods, while also providing well-calibrated uncertainty estimates that reliably indicate when and where predictions degrade. The results demonstrate a robust, data-driven framework for flow field reconstruction with meaningful uncertainty quantification and offer practical guidance for experimental design in periodic flows.