> The story of Newton sitting under an apple tree may be legend, but the most exciting moments in the history of science share a pattern: someone captured nature's equations from observational data. Newton distilled universal gravitation from planetary orbits, Kepler found elliptical orbits from Mars's positions, and Maxwell wrote four equations from electromagnetic experiments. Each time, the human brain — a "pattern recognition engine" — found concise mathematical structure within chaotic data.
> Today's paper asks an audacious question: Can AI play that role?
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The Dilemma of SINDy
In 2016, Brunton, Proctor, and Kutz proposed SINDy (Sparse Identification of Nonlinear Dynamics) — a method for automatically discovering equations of motion from data. Its core idea is elegant:
1. Assume the system's dynamics can be described as 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.
It's like being asked to pick a few pages from a mathematical dictionary and assemble an article that accurately describes the data, with as few pages as possible.
SINDy has a fatal flaw: you must prepare the "candidate function library" in advance. This means guessing beforehand what kinds of nonlinearities the system might involve — polynomial, trigonometric, exponential, or some complex combination. For many novel systems, if you don't know the underlying physical structure, SINDy has no starting point.
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AutoSINDy's Three-Step Recipe
AutoSINDy creatively combines two worlds:
- Symbolic regression (PySR): exploration magic that automatically discovers function forms
- SINDy: the wisdom of sparse selection — picking the simplest candidate from a pool
- Standard SINDy (with augmented library): succeeds only when the candidate library happens to contain the correct functions. For unconventional nonlinearities, guessing the library correctly without prior human knowledge is nearly impossible.
- Pure symbolic regression (PySR alone): without sparsity constraints, it tends to produce overly complex, numerically unstable equations, especially on noisy data.
- Climate science: discovering new equations of atmospheric circulation from observational data
- Neuroscience: finding neural population dynamics from EEG data
- Materials science: summarizing how matter responds to external 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 ID: 2605.09696
- Published: May 10, 2026
- Categories: cs.LG, cs.NE, cs.SC
- Method: PySR symbolic regression + SINDy sparse identification
It operates in three stages:
Stage 1: Symbolic Regression's "Free Exploration"
Multiple bootstrapped chunks of the observational data are sampled, and each chunk is fed independently into PySR, a state-of-the-art symbolic regression engine, to discover candidate function forms. Since each round sees a slightly different subset of data, it may discover different functions — some may be noise, some may be real structure.Stage 2: Curation and Expansion
This is the cleverest part. PySR's output can be a messy pile of function combinations. AutoSINDy cleans them up via collinearity analysis — if two candidate functions are highly correlated on the data (e.g., x² and (x+1)² over a particular data range), they are merged or deduplicated. The remaining functions are then decomposed and expanded — for example, from "sin(x)", variants like "sin(2x)" and "sin(x²)" are also generated as candidates.Stage 3: Sparse Selection
SINDy's core algorithm then selects the sparsest combination from the curated candidate pool. Because the earlier steps have greatly improved candidate quality, SINDy performs far better in this stage than in its original form.---
92.8% — What Does This Number 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.
Benchmarks for comparison:
AutoSINDy stacks the strengths of both: symbolic regression ensures "no possibility is missed," while SINDy ensures "no unnecessary complexity."
More importantly, it maintains a high recovery rate even under high noise (signal-to-noise ratio as low as 20:1), and shows remarkable generalization — the recovered equations stay accurate on trajectories beyond the training data range.
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From "Guessing" to "Automated Discovery" — An Evolution of the Scientific Method
This work recalls Feynman's description of the scientific method in the *Feynman Lectures on Physics*: "First we guess a formula, then we compute its consequences. If the computed consequences agree with 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 — "structured" means candidate functions must satisfy basic mathematical consistency (e.g., collinearity constraints), and "search" means it doesn't require humans to pre-specify function families.
This opens new possibilities across many fields:
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A Feynman-Style Reflection
At MIT, Feynman once complained: "It took physicists fifty years to figure out superconductivity — not because the math was too hard, but because no one guessed the basic physical picture correctly."
"AutoSINDy makes me feel that in the future, we may no longer need to 'guess.' You collect data, the AI gives you a list of candidate equations, and you pick the physically plausible one and verify it experimentally.
Of course, AI doesn't understand physics — it only understands finding patterns in data. But sometimes the pattern itself *is* the physics. A Lorenz attractor is just a set of numbers to the AI, yet it 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 is not a person — but a person and an AI together, the former providing physical intuition, the latter mathematical discovery."
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*Paper Information*