AutoSINDy: Letting AI Play Newton — Writing Physics Equations from Data with 92.8% Accuracy
The story of Newton under the apple tree may be legend, but science's most thrilling moments share a pattern: someone captures nature's equations from observational data. Newton distilled universal gravitation from planetary trajectories, Kepler found elliptical orbits from Mars positions, and Maxwell wrote four equations from electromagnetic experiments. This post discusses a paper that asks a bold question: can AI be that person?
The Dilemma of SINDy
In 2016, Brunton, Proctor, and Kutz proposed SINDy (Sparse Identification of Nonlinear Dynamics), a method to automatically discover equations of motion from data. Its core idea is elegant:
1. Assume the system's dynamics can be described by a sparse linear combination of candidate functions (e.g., x, x², x³, xy, sin(x)...) 2. Use an optimization algorithm to find the most "economical" combination — if two terms suffice, don't use three.
SINDy has a fatal flaw: you must prepare the "candidate function library" first. This means guessing in advance which types of nonlinearity the system might involve — polynomial, trigonometric, exponential, or some complex combination. For genuinely novel systems where the underlying physical structure is unknown, SINDy has no way to start.
AutoSINDy's Three-Step Recipe
AutoSINDy creatively combines two worlds:
- The exploration magic of symbolic regression (PySR) — automatically discovering function forms
- The sparse-selection wisdom of SINDy — picking the simplest candidate from the pool
- Standard SINDy (with augmented library): succeeds only if the library happens to contain the correct functions. For unconventional nonlinearities, guessing the library without prior knowledge is nearly impossible.
- Pure symbolic regression (standalone PySR): without sparsity constraints, tends to produce overly complex, numerically unstable equations, especially with noisy data.
- Climate science: discovering new atmospheric circulation equations from observational data
- Neuroscience: finding neural population dynamics from EEG data
- Materials science: summarizing matter's response to forces from experiments
- Finance: discovering volatility evolution rules from market data
- Title: Discovery of Nonlinear Dynamics with Automated Basis Function Generation
- Authors: Mohammad Amin Basiri, Charles Nicholson
- arXiv: 2605.09696
- Published: May 10, 2026
- Categories: cs.LG, cs.NE, cs.SC
- Method: PySR symbolic regression + SINDy sparse identification
Stage 1: Free Exploration via Symbolic Regression
Multiple bootstrapped chunks of the observational data are fed to PySR, a state-of-the-art symbolic regression engine, with each round independently discovering candidate function forms. Since each round sees slightly different data subsets, it may find different functions — some noise, some true structure.
Stage 2: Curation and Expansion
PySR output can be a messy pile of function combinations. AutoSINDy cleans up via colinearity analysis — if two candidate functions are highly correlated on the data (e.g., x² and (x+1)² within a certain range), they are merged or deduplicated. Remaining functions are then decomposed and expanded — e.g., from "sin(x)", generate "sin(2x)" and "sin(x²)" as additional candidates.
Stage 3: Sparse Screening
SINDy's core algorithm selects the sparsest combination from the curated candidate pool. Because earlier steps have greatly improved pool quality, SINDy far outperforms its original version at this stage.
What Does 92.8% Mean?
On standard nonlinear systems (Lorenz attractor, Van der Pol oscillator, Duffing equation, etc.), AutoSINDy recovered the exact ground-truth equations in 92.8% of trials.
Comparison with baselines:
AutoSINDy stacks both advantages: symbolic regression ensures "no possibility is missed," while SINDy ensures "no over-complication."
Moreover, it maintains a high recovery rate even under high noise (signal-to-noise ratio as low as 20:1), plus strong generalization — recovered equations stay accurate on trajectories beyond the training data range.
From "Guessing" to "Automatic Discovery"
This work evokes Feynman's description of the scientific method: "First we guess a formula, then we compute its consequences. If the computed consequences match experiment, we consider the guess right."
AutoSINDy automates the hardest step — "guessing the formula." And it doesn't guess randomly: it searches in a structured space, where candidates must satisfy basic mathematical consistency (colinearity constraints), without humans pre-specifying function families.
This opens new possibilities across fields:
Closing Thoughts
AI doesn't understand physics — it only finds patterns in data. But sometimes the pattern *is* the physics. A Lorenz attractor is just a set of numbers to AI, yet AutoSINDy recovered three lines of differential equations:
ẋ = σ(y-x), ẏ = x(ρ-z)-y, ż = xy-βz
— exactly what Lorenz wrote on the blackboard in 1963. Perhaps the future Newton isn't a person, but a person and an AI: the former providing physical intuition, the latter providing mathematical discovery.