🌀 nonlinear sciences

Machine-Precision Prediction of Low-Dimensional Chaotic Systems from Noise-Free Data

This paper demonstrates that using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic allows for machine-precision prediction of low-dimensional chaotic systems from noise-free data, effectively solving the forecasting problem by significantly outperforming prior methods and standard numerical solvers.

Original authors: Christof Schötz, Niklas Boers

Published 2026-08-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Christof Schötz, Niklas Boers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict the path of a leaf swirling in a storm. In the world of science, this is known as a "chaotic system." These are complex machines—like weather patterns, flowing water, or even the beating of a heart—where tiny changes at the start lead to wildly different outcomes later on. It's the famous "butterfly effect": a butterfly flapping its wings in Brazil could theoretically cause a tornado in Texas. Because of this extreme sensitivity, predicting these systems is notoriously difficult; usually, the longer you try to forecast, the more your prediction drifts away from reality, like a GPS that slowly loses signal until it's pointing you in the wrong country.

For decades, scientists have used a specific, simplified model called the Lorenz-63 system to test their prediction tools. Think of it as the "training wheels" of chaos theory—a three-dimensional mathematical playground where researchers can see if their new algorithms are actually learning the rules of the game or just guessing. The big question has always been: Can we learn these rules perfectly from data alone, without needing to know the underlying physics equations? Until now, the best attempts could only keep up with the system for a short while before the prediction fell apart.

This study introduces a new method called PolyProp (short for Polynomial Propagator) that changes the game. The researchers found that if you have perfect, noise-free data, you can predict the future of these chaotic systems with almost impossible precision—so precise that it matches the accuracy of the super-computers that generated the data in the first place.

Here is how they did it and what they found:

The Magic of High-Degree Polynomials
Most machine learning models try to find a simple curve to fit the data, like drawing a straight line through a scatter of dots. But chaotic systems are messy; they twist and turn in ways a simple line can't capture. PolyProp takes a different approach. It tries to fit a "super-curve" made of high-degree polynomials. Imagine trying to describe the shape of a rollercoaster. A simple line fails, but a complex, twisting mathematical formula can trace every loop and drop perfectly.

The catch? These formulas are incredibly sensitive to tiny errors, like trying to balance a house of cards in a hurricane. If your math isn't perfect, the whole thing collapses. To solve this, the authors didn't just use standard computer math (which is like using a ruler with millimeter markings). They used 512-bit arithmetic, which is like using a ruler with markings so fine they are smaller than an atom. This allowed them to calculate the complex polynomial coefficients without losing any precision.

The Results: Breaking the Record
The team tested PolyProp on the Lorenz-63 system. Previous record-holders could predict the system's behavior for about 13 "Lyapunov times" (a unit of time that measures how fast chaos grows). In plain English, that's how long the prediction stays close to the truth before the butterfly effect takes over.

PolyProp shattered this record.

In its standard setup, it achieved a valid prediction time of 36 Lyapunov times.
With the most favorable settings, it reached up to 105 Lyapunov times.

To put that in perspective, the paper notes that this is an improvement of a factor of 10 billion over the previous state-of-the-art. The method was so accurate that it matched the performance of the numerical solvers that actually know the system's governing equations. In other words, by just looking at the data and using high-precision math, PolyProp learned the system's secrets better than anyone thought possible.

Does it work on other systems?
The researchers didn't stop there. They tested PolyProp on:

Thomas' Cyclically Symmetric Attractor: A system that is even more complex and doesn't follow simple polynomial rules (it uses sine waves). Even here, PolyProp matched the precision of the best possible solvers, reaching about 38 Lyapunov times.
The Lorenz-96 System: A model used to simulate weather patterns that is much larger and more complex. PolyProp successfully handled these higher dimensions, proving the method scales up.

The Big Caveat: The "Noise" Problem
While the results are stunning, the paper is very clear about one major limitation: Noise.
The "perfect" results above rely on noise-free data. In the real world, measurements are never perfect; they have static, errors, and fuzziness (like a radio with static). The authors explain that if you add noise to the data, the prediction accuracy drops rapidly, just like it does for other methods.

The study suggests that for noise-free data, the problem of learning low-dimensional chaotic systems is effectively solved. However, for real-world applications where data is messy, the challenge shifts. The paper argues that the solution isn't just better math or higher precision, but better ways to clean the data (denoising) before trying to predict.

In Summary
This paper shows that if you can get clean, precise data, you don't need to know the physics equations to predict chaotic systems. By using a clever combination of high-degree polynomials and super-precise math, you can learn the system's behavior so well that you can forecast it for incredibly long periods—far longer than anyone thought possible. It's a massive leap forward for the theory of chaos, even if the real-world application still needs to figure out how to handle the static on the radio.

Technical Summary: Machine-Precision Prediction of Low-Dimensional Chaotic Systems from Noise-Free Data

Problem Statement

Predicting the behavior of chaotic dynamical systems is a fundamental challenge in science due to their extreme sensitivity to initial conditions. While data-driven, system-agnostic machine learning (ML) methods have emerged as promising tools for forecasting such dynamics, their performance is typically limited. Standard benchmarks, such as the Lorenz-63 (L63) model, have historically shown that state-of-the-art methods (including Reservoir Computing, LSTMs, and SINDy) achieve valid prediction times (VPT) of approximately 13 Lyapunov times at best. These methods often rely on regularization to handle noise or numerical instability, which introduces bias and limits precision. The central problem addressed is whether it is possible to learn the dynamics of low-dimensional chaotic systems from noise-free data with an accuracy that matches or exceeds that of standard numerical Ordinary Differential Equation (ODE) solvers utilizing the system's governing equations.

Methodology: PolyProp

The authors introduce PolyProp (Polynomial Propagator), a system-agnostic method that learns and predicts chaotic dynamics using ordinary least squares (OLS) regression on high-degree polynomial features. The core components of the methodology include:

Propagator Estimation: Instead of learning the vector field $f(u)$ directly (as in SINDy), PolyProp estimates the propagator map $\Phi_{\Delta t}$ , which maps a state $u(t)$ to $u(t+\Delta t)$ .
High-Degree Polynomial Features: The method constructs a feature vector using multivariate monomials up to a maximal degree $p$ . For a system of dimension $d$ , the number of features is $D = \binom{d+p}{d}$ . The study utilizes degrees up to $p=25$ for certain systems.
Multi-Precision Arithmetic: A critical innovation is the use of 512-bit arithmetic (via MPFR and MPLAPACK libraries) for solving the linear systems required for OLS. Standard 64-bit double precision is insufficient for high-degree polynomial fitting due to the severe ill-conditioning of the resulting matrices. The 512-bit precision ensures that numerical errors do not dominate the solution, allowing the method to exploit the full precision of noise-free data.
No Regularization: Unlike many ML approaches, PolyProp does not employ regularization (e.g., Ridge regression). The authors argue that in the absence of noise, regularization introduces unnecessary bias that degrades accuracy.
Experimental Setup: The method is tested on three benchmark systems:
- Lorenz-63 (L63): A 3D polynomial system.
- Thomas' Cyclically Symmetric Attractor (TCSA): A 3D non-polynomial system involving sine functions.
- Lorenz-96 (L96): A spatiotemporally chaotic system with dimensions $d=5$ to $9$.
- Ground truth data is generated using high-precision (512-bit) RK4 ODE solvers, then stored at 64-bit precision to simulate realistic data storage constraints.

Key Results

1. Unprecedented Valid Prediction Times (VPT)

On the Lorenz-63 system, PolyProp achieves a VPT of 36 Lyapunov times in the default setup (512-bit method, 64-bit data), and up to 105 Lyapunov times with favorable precision configurations. This represents a dramatic improvement over the previous state-of-the-art of ~13 Lyapunov times.

Error Magnitude: In the first Lyapunov time unit, PolyProp achieves a normalized root mean squared error (nRMSE) of $10^{-16}$ to $10^{-15}$ , comparable to 64-bit machine precision ( $2^{-52}$ ). This is an improvement of a factor of $10^9$ over prior data-driven methods.

2. Matching Numerical Solvers

PolyProp's performance is statistically indistinguishable from, and in some cases slightly exceeds, that of numerical ODE solvers (RK4) initialized with the same precision.

When trained on 64-bit rounded data, a 512-bit PolyProp forecast diverges from the ground truth at the same rate as a 64-bit RK4 solver.
This indicates that the forecast error is dominated by the initial condition error (the rounding of the starting state) rather than the dynamics error (the error in the learned model). PolyProp effectively learns the dynamics so perfectly that the only remaining limitation is the precision of the initial state.

3. Generalization to Non-Polynomial and Higher-Dimensional Systems

TCSA: Despite the vector field containing non-polynomial sine functions, PolyProp (using high-degree polynomials) achieves a VPT of 38.0 Lyapunov times, matching the performance of a multi-precision ODE solver.
Lorenz-96: The method successfully extends to higher dimensions ( $d=5$ to $9$), achieving single-precision (32-bit) machine accuracy when trained on single-precision data, matching the accuracy of a double-precision ODE solver initialized with single-precision conditions.

4. Noise and Precision

The study treats finite numerical precision as a form of deterministic noise. It demonstrates that PolyProp trained on reduced-precision data yields forecasts indistinguishable from an ODE solver starting from the same reduced-precision initial condition. When applied to stochastic noise, PolyProp achieves the optimal forecast accuracy possible for non-denoising methods, again matching the ODE solver initialized with the same noisy state.

Significance and Claims

The paper claims that forecasting low-dimensional chaotic systems from noise-free data is effectively a solved problem.

Overcoming the "Black Box": The results suggest that the lack of knowledge of the governing equations is not a fundamental barrier to high-precision prediction. With sufficient data, suitable learning algorithms, and adequate numerical precision, a data-driven approach can perform at least as well as a physics-based integrator.
Re-evaluating Benchmarks: The authors argue that standard benchmarks like the noise-free Lorenz-63 model are no longer adequate for distinguishing between methods, as they have become "trivial" under these conditions. The previous limits on prediction were likely due to insufficient numerical precision and the use of regularization, rather than an inherent inability of data-driven methods to capture chaotic dynamics.
Limitations: The authors explicitly note that while the problem is solved for noise-free data, real-world applications involve measurement noise. In such cases, the primary obstacle remains noise, and future research must focus on methods that inherently integrate denoising capabilities, as higher precision alone cannot overcome stochastic noise.

In summary, the paper demonstrates that the combination of high-degree polynomial regression and multi-precision arithmetic allows for the machine-precision prediction of chaotic systems, fundamentally changing the understanding of the limits of data-driven forecasting in low-dimensional settings.