Aug 2026· Proceedings of the 32nd ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.2· pp. 10854-10865· 0 citations· 30 references
Abstract
Discovering governing Partial Differential Equations (PDEs) from observational data is a fundamental challenge in AI for Science. While Symbolic Regression (SR) dominates this task, existing token-level methods trigger a combinatorial explosion of search spaces, frequently yielding mathematically valid yet physically inconsistent candidates. To overcome this, we propose Modular Symbolic Regression with Physics Priors (MSR-PP), a knowledge-guided framework that conceptualizes one-dimensional PDEs as structured compositions of semantic modules (e.g., convection, diffusion) rather than random symbol sequences. MSR-PP employs an RL-optimized LSTM agent to sequentially construct equation skeletons based on syntactic and semantic dependencies, utilizing a modular search strategy to significantly prune the search space. Addressing the critical scarcity of standardized, high-fidelity benchmarks for 1D PDEs, we constructed an extended dataset for rigorous evaluation. Extensive experiments demonstrate that MSR-PP outperforms state-of-the-art baselines. Real-world validation on the NGSIM US101 traffic dataset shows MSR-PP successfully identifies a convection-diffusion model featuring a critical second-order term. Notably, the negative coefficient of this term (-u_xx) aligns with theoretical models, accurately capturing the string instability and wave-amplifying dynamics characteristic of stop-and-go traffic. Moreover, the discovered equation exhibits robust out-of-distribution (OOD) generalization on the unseen NGSIM I-80 dataset, underscoring MSR-PP's capability to uncover physically meaningful laws. Datasets: https://github.com/jinyangdu5/MSR-PP.
EvoPINN is proposed, an agentic framework that reformulates PINN development from labor-intensive manual design into a rigorous, execution-grounded algorithm discovery problem, and autonomously invented SLRC-PINN, a novel architecture whose performance gains persist under rigorous parameter-matched comparisons.
Peng Yin, Kai Li, Yifan Zhang et al.· arXiv.org· 0 citations
Symbolic regression (SR), the automated discovery of mathematical expressions from data, is a cornerstone of scientific inquiry. However, it is often hindered by the combinatorial explosion of the search space and a tendency to overfit. Popular methods, rooted in genetic programming, explore this space syntactically, often yielding overly complex, uninterpretable models. This paper introduces IdeaSearchFitter, a framework that employs Large Language Models (LLMs) as semantic operators within an evolutionary search. By generating candidate expressions guided by natural-language rationales, our method biases discovery towards models that are not only accurate but also conceptually coherent and interpretable. We demonstrate IdeaSearchFitter's efficacy across diverse challenges: it achieves competitive, noise-robust performance on the Feynman Symbolic Regression Database (FSReD), outperforming several strong baselines; discovers mechanistically aligned models with good accuracy-complexity trade-offs on real-world data; and derives compact, physically-motivated parametrizations for Parton Distribution Functions in a frontier high-energy physics application. IdeaSearchFitter is a specialized module within our broader iterated agent framework, IdeaSearch, which is publicly available at \href{https://www.ideasearch.cn/}{https://www.ideasearch.cn/}.
Zhuo-Yang Song, Ze-Yu Cai, Shu-Tao Zhang et al.· Communications in Theoretica...· 0 citations
Meta-SPINN works both as a direct predictor for unseen tasks and as a task-aware initializer for subsequent single-instance residual-guided refinement, providing reusable predictions together with an interpretable visualization of how solution geometry changes across a parameter family.
This paper proposes Kastor, a comprehensive methodology to adapt a deterministic physics foundation model into a highly efficient and accurate generative surrogate, and introduces a two-stage inference scheme that combines a large-stride causal auto-regressive model with a non-causal temporal super-resolution network, significantly reducing error accumulation while minimizing computational cost.
Guillaume Couairon, Alexis Jacq, Yu-Han Wu et al.· 0 citations
Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained transformers mapping synthetic ODE trajectories to equations offer interpretable alternatives, promising transfer without system-specific equation knowledge. Transferring them reliably to high-dimensional physical data, however, remains an open challenge. We develop a verifier-guided (VG) workflow around ODEFormer as a symbolic backbone, using dynamical and physical-admissibility criteria to select from a multi-trajectory candidate equation pool, enabling transfer. On canonical Van der Pol oscillators, VG outperforms the original ODEFormer workflow across held-out initial conditions. We then address vortex shedding, a phenomenon occurring in atmospheric and plasma systems of societal relevance, through coordinate reduction and symbolic discovery at fixed and varying Reynolds numbers. VG discovers fixed-parameter reduced-order equations that recover the fundamental shedding oscillator and higher harmonics without a wake-specific candidate library or prescribed Navier-Stokes structure, while the cross-parameter model generalizes to withheld regimes. Reconstruction fidelity alone did not determine symbolic discoverability, highlighting the importance of compatibility between latent dynamics and the backbone's pretraining distribution. This work establishes a verifier-guided neural-to-symbolic methodology for interpretable and physically auditable forecasting in the natural sciences.
Partial differential equation (PDE) discovery aims to identify from data the governing law of a physical system. Constituting a cornerstone of scientific advancement, it has become during the past decade a major line of research in the rapidly evolving field of Physics-informed Machine Learning (PiML). Among the remaining open problems to address in this domain, the post-hoc evaluation of discovered PDEs raises the particular difficulty of being multifaceted. Indeed, it requires jointly considering predictive accuracy, physical consistency, interpretability, and out-of-distribution generalization capacity. Given that some of these properties are conflicting, it is worth noting that the wide range of existing evaluation metrics only partially address the overall problem, potentially leading to overly interpreted conclusions about the validity of a presumed new physical theory. From an abundant literature spanning machine learning, numerical analysis, information theory or symbolic regression, we propose, to our knowledge, the first taxonomy of PDE evaluation metrics, and discuss their advantages and limitations in depth. Based on the observation that evaluation is often achieved on a case-by-case basis and that a universally accepted methodology remains elusive, we further provide recommendations with the aim of promoting standardized and reliable practices, before sketching promising future lines of research in this field. We argue that this paper is intended both for ML experts who design new PDE discovery algorithms and for users of these methods aiming, in real applications, to discover and validate well-founded scientific laws.
