One of the deepest ambitions of computational science is to watch a complex system, a turbulent fluid, a chemical reaction, an ecosystem, and extract the mathematical rules that govern its behavior. This is “equation discovery” or “symbolic regression,” and it has been pursued for decades. The challenge is that real-world data is noisy, multi-scale, and incomplete. Classical methods like SINDy (Sparse Identification of Nonlinear Dynamics) fail in the presence of even modest noise. Neural network approaches, while robust, produce black boxes: they can predict but cannot explain.
A team at the Shenyang Institute of Automation, part of the Chinese Academy of Sciences, has now developed a framework that combines the best of both worlds. Called PK-MCL (Physics-Koopman Multi-scale Contrastive Learning), it can extract clean, physically interpretable governing equations from data contaminated by up to 10% noise, including missing spatial regions and temporal gaps. The work was published July 13 in Nature Communications.
“Our framework shifts the problem from static curve-fitting to constrained dynamical inference,” said corresponding author Xiaofeng Zhou. “The equation you recover must not only fit the training data but also produce stable, physically meaningful predictions over long time horizons.”
Three modules
PK-MCL integrates three components. The first is a multi-scale Koopman neural operator that decomposes the input field into different frequency bands using Fourier transforms, essentially, separating fast and slow dynamics, and evolves each band linearly in a learned latent space. This spectral decomposition is essential for systems in which different physical processes operate on different timescales, such as chemical reactions embedded in a flowing fluid.
The second component is a physics-guided sparse projection that constrains the output equation to be composed of a small number of physically meaningful terms from a pre-specified library, polynomials, spatial derivatives, and other interpretable building blocks. This is embedded directly into the training process, not applied as a post-hoc correction, so the neural network is explicitly guided toward identifying compact equations from the start.
The third component is a multi-view consistency regularization, borrowed from self-supervised learning (the BYOL architecture), which forces the model to produce invariant representations across different perturbations of the input, masking, noise injection, temporal dropout. This dramatically improves robustness.
The three components are trained jointly through a single loss function that balances prediction accuracy, equation sparsity, and representation consistency.
Benchmarks
The researchers tested PK-MCL on a battery of canonical systems: the Burgers equation (a model of nonlinear advection-diffusion), the two-dimensional FitzHugh-Nagumo reaction-diffusion system, and the two-dimensional Navier-Stokes vorticity equation. In all cases, PK-MCL recovered the correct governing equations with high fidelity, even under 10% measurement noise, conditions under which classical SINDy and its variants break down completely.
The framework also demonstrated stable long-horizon predictions, maintaining accuracy over hundreds of time steps, and generalized to unseen initial conditions and operating regimes. On the Navier-Stokes benchmark, it preserved the large-scale vortex structures and the energy cascade that baselines failed to capture.
Beyond synthetic benchmarks, the team validated PK-MCL on real sensor data from an industrial grinding and classification circuit, a mineral-processing system with eight measured variables, sensor noise, and missing samples. The framework extracted physically meaningful relationships between variables that matched known plant behavior.
What makes it interpretable
Unlike a standard neural network, PK-MCL outputs a sparse set of coefficients that map directly to interpretable mathematical terms: advection, diffusion, reaction rates. The user doesn’t just get a prediction, they get an equation. And the spectral decomposition reveals which frequency bands govern which phenomena, providing additional physical insight.
Several caveats apply. The method requires a pre-specified library of candidate terms; if the true governing equation uses functions not in the library, recovery will fail. The benchmarks are limited to one- and two-dimensional systems; three-dimensional problems would be computationally demanding. And the framework has not yet been tested on systems with discontinuities or stochastic noise. The paper is published as an advance manuscript, before final copy-editing.
Nevertheless, PK-MCL represents a significant step toward the goal of automated scientific discovery: feeding raw data into a machine and getting back a concise, human-readable equation that advances understanding of the underlying physics. For fields from climate science to systems biology to engineering, that capability could transform how models are built.
Source: Jia, D., Li, S., Zuo, X. et al. “From data chaos to physically interpretable deterministic mapping.” Nature Communications (2026). DOI: 10.1038/s41467-026-75164-9