[Paper Notes] From Kepler to Newton: Explainable AI for Science
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TL;DR
“From Kepler to Newton: Explainable AI for Science” uses a historically elegant test case to argue for Explainable AI in scientific discovery. The authors ask: if modern AI had access only to Tycho Brahe and Kepler’s Mars observations, could it rediscover pieces of Kepler’s planetary laws and Newton’s inverse-square gravitation? Their answer is yes, but with an important caveat. Black-box neural networks can fit and augment sparse observations, symbolic regression can expose compact equations, but humans still need to interpret variables, invent concepts such as acceleration and force, and prevent the equations from being over-generalized.
Paper Info
The paper is “From Kepler to Newton: Explainable AI for Science” by Zelong Li, Jianchao Ji, and Yongfeng Zhang from Rutgers University. It appeared at the 2nd AI4Science Workshop at ICML 2022 and is available as arXiv:2111.12210. I reviewed the arXiv PDF and kept a local copy at assets/papers/2111.12210.pdf.
1. Motivation
The paper starts from the classic scientific loop:
observation -> question -> hypothesis -> prediction -> experiment and test
This is the hypothetico-deductive paradigm. It has served science well, but modern science increasingly has too much data for researchers to manually inspect every possible pattern. The authors frame AI as a way to help with this bottleneck, especially in the step where a scientist proposes useful hypotheses from observations.
Their proposed alternative is an Explainable AI-based, hypothesis-free discovery loop:
observation -> question -> model learning -> model interpretation -> prediction -> experiment and test
The key substitution is not simply “use AI.” It is more specific: use black-box models for accurate prediction and data augmentation, then use white-box explanation methods to translate the learned model into human-readable equations or rules. In the authors’ framing, AI can generate candidate hypotheses, but human scientists still decide what those hypotheses mean.
2. Why Kepler and Newton?
The historical analogy is clean:
- Tycho Brahe represents observation: careful astronomical measurements.
- Johannes Kepler represents model learning: discovering regularities in planetary motion.
- Isaac Newton represents explanation: explaining those regularities through force, acceleration, and gravitation.
The paper asks what this pipeline would look like if part of Kepler and Newton’s work were assisted by modern AI. To keep the setting historically constrained, the experiments use Mars observation data from Kepler’s Astronomia Nova, not modern high-precision planetary data. That choice matters because the paper wants to test whether AI can rediscover insight from the kind of limited but high-quality data available around the scientific revolution.
3. Model Learning: Black-Box Prediction and Data Augmentation
The first model is deliberately simple: a three-layer MLP with hidden size 100. The authors use the same neural-network structure across experiments instead of hand-designing a physics-aware architecture.
For Kepler’s first law, the network learns a function:
\[r = NN(\theta)\]where (r) is the Sun-Mars distance and (\theta) is Mars’ angular position in heliocentric ecliptic coordinates. The original dataset has only 28 observations, so the neural network is used both for smooth fitting and for generating 1,000 augmented samples.
This is the “black-box AI” part of the paper. It can make useful predictions, and it can turn sparse historical data into a denser function-like object. But by itself, the MLP does not tell us why Mars moves as it does. A nested set of learned weights is not the same as a scientific explanation.
4. Model Interpretation: Symbolic Regression and Kepler’s First Law
The explanation step uses symbolic regression, implemented with TuringBot. The goal is to translate the black-box function (r = NN(\theta)) into an explicit expression (r = f(\theta)).
Using cosine, addition, multiplication, and division as base operations, symbolic regression finds a compact expression:
\[r = \frac{1.51033}{1 + 0.0927177 \cos(\theta + 0.544536)}\]This matches the polar form of an ellipse:
\[r = \frac{l}{1 + \epsilon \cos(\theta)}\]So the AI pipeline recovers Kepler’s first law in an interpretable form: Mars follows an elliptical orbit with the Sun at a focus. Even better, the coefficient (0.0927177) can be read as Mars’ eccentricity. The paper reports that this is within 0.1% of Kepler’s own value and about 0.7% from the modern value, reasonable given that the experiment intentionally uses 400-year-old data.
The phase shift (\theta + 0.544536) also has a physical interpretation. It suggests that Mars reaches perihelion around August, consistent with historical records of close Mars oppositions.
5. From Kepler Toward Newton
The second experiment adds time. The authors train another MLP:
\[\theta = NN(t)\]where (t) is time normalized into one Mars orbital period. This is useful because the exact inverse relationship from time to angle is hard to express directly; the paper notes that the known orbital mechanics equation gives (t) as a function of (\theta), but (\theta) as a function of (t) is not easy to write in closed form.
Because the neural model is smooth, it can still support analysis. The authors estimate angular velocity (\omega) by finite differences and combine it with (r), then ask symbolic regression to search for relations among (r), (r^2), (r^3), (\omega), (\omega^2), and (\omega^3).
The compact discovered rule is:
\[r^3 \omega^2 = 0.000298491 \ \mathrm{AU^3 day^{-2}}\]Modern physics recognizes the same structure as:
\[r^3 \omega^2 = GM\]The reported relative error is about 0.8% compared with the modern value in AU and days. This is the paper’s strongest example of the model-learning plus model-interpretation workflow: neural networks make the time-angle relationship usable, symbolic regression identifies a simple invariant, and that invariant points toward a real physical law.
6. The Human Role: Meaning Is Not Automatic
The most interesting part of the paper is not just that symbolic regression finds equations. It is the authors’ insistence that equations are not automatically science.
For example, symbolic regression can output:
\[r^3 \omega^2 = c\]But to reach Newtonian gravitation, a human still has to recognize that (a = r\omega^2) is centripetal acceleration, then reorganize the expression as:
\[a \propto \frac{1}{r^2}\]and finally connect force to acceleration through (F = ma), yielding:
\[F \propto \frac{1}{r^2}\]This is the conceptual leap. AI can narrow the hypothesis search space and expose useful symbolic structure, but the meaning of “acceleration,” “force,” and “gravity” is assigned through human scientific understanding.
The paper also warns against over-reading the result. The Mars-only relationship (r^3\omega^2=c) is not itself Kepler’s third law, because Kepler’s third law is a cross-planet statement about orbital period and mean distance. Still, it points in the right direction: if one considers (\bar{\omega}=2\pi/T), the expression suggests a relation between (\bar{r}^3) and (T^2), which could guide the search once data from more planets is included.
7. Strengths
- The historical case study is unusually readable. It gives the paper a clear scientific-discovery pipeline instead of only a machine-learning benchmark.
- The division between prediction and explanation is sharp. The authors do not pretend that a high-accuracy MLP is a scientific theory.
- Symbolic regression is used in a natural role: translating a fitted model into candidate laws.
- The paper is careful about the human scientist’s role. It treats AI-generated equations as hypotheses and search-space reducers, not as self-sufficient knowledge.
8. Limitations
- The experiment is more a demonstration than a full discovery engine. The search space is still strongly shaped by the authors’ chosen variables and base functions.
- The physical domain is clean and low-dimensional. Real AI-for-science settings may involve noisy, high-dimensional, partially observed systems where symbolic regression is much harder.
- The paper depends on human interpretation at key points, which is philosophically honest but also means the AI pipeline is not autonomous.
- The singularity discussion is provocative but less technically grounded than the Kepler/Newton experiments. The most durable contribution is the concrete XAI-for-science workflow.
Takeaways
This paper is valuable because it makes a precise distinction: prediction is not explanation, and explanation is not yet meaning. Black-box models can be powerful scientific instruments, especially for fitting sparse data and creating differentiable approximations. Symbolic regression can turn those instruments into candidate equations. But science still needs humans to judge scope, invent concepts, connect equations to mechanisms, and decide when an apparent rule is real knowledge rather than an overfit pattern.
For AI-for-science, that is a useful stance. The right goal is not to replace the scientist with an inscrutable oracle. It is to build systems that accelerate the observation-to-hypothesis step while keeping the resulting knowledge legible enough for humans to inspect, challenge, and extend.