Data-driven discovery of governing differential equations across physical systems

Researchers are making advances in data-driven differential equation discovery and probabilistic forecasting, with new approaches emerging to improve the accuracy and reliability of these methods.

Differential equations play a critical role in scientific discovery, providing a mathematical framework to describe physical phenomena^[1]. Data-driven differential equation discovery has attracted increasing attention as a promising alternative to traditional first principles. The field has expanded rapidly, with AI-based approaches emerging to infer governing laws directly from experimental or simulated data. A review on arxiv.org^[1] proposes a problem-oriented perspective on data-driven differential equation discovery, introducing a two-dimensional phase diagram of equation discoverability. This diagram shows how the field has moved from simple to complex governing laws. The review also presents the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. Meanwhile, another study on arxiv.org^[2] compares generative models and ensembles of deterministic models with stochasticity injected for generating probabilistic forecasts of physical systems. The study found that CRPS-trained ensembles typically achieve more reliable uncertainties than generative models, and offer faster inference. However, generative models trained in ambient rather than latent space exhibit comparable coverage to CRPS-trained ensembles^[2].