AI in physics and maths¶
What Has a Foundation Model Found? Using Inductive Bias to Probe for World Model¶
{2507.06952} What Has a Foundation Model Found? Using Inductive Bias to Probe for World Models, Keyon Vafa, Peter G. Chang, Ashesh Rambachan, Sendhil Mullainathan (2025)
Foundation models are premised on the idea that sequence prediction can uncover deeper domain understanding, much like how Kepler’s predictions of planetary motion later led to the discovery of Newtonian mechanics. However, evaluating whether these models truly capture deeper structure remains a challenge. We develop a technique for evaluating foundation models that examines how they adapt to synthetic datasets generated from some postulated world model. Our technique measures whether the foundation model’s inductive bias aligns with the world model, and so we refer to it as an inductive bias probe. Across multiple domains, we find that foundation models can excel at their training tasks yet fail to develop inductive biases towards the underlying world model when adapted to new tasks. We particularly find that foundation models trained on orbital trajectories consistently fail to apply Newtonian mechanics when adapted to new physics tasks. Further analysis reveals that these models behave as if they develop task-specific heuristics that fail to generalize.
Quotes
Taken together, our results provide a direction for understanding the deficiencies of foundation models: if a model’s inductive bias isn’t toward a known model of reality, what is it toward? We explore this question by examining whether these foundation models have alternative inductive biases. Our analysis reveals that these models instead behave as if they develop task-specific heuristics that fail to generalize. For physics, rather than learning one universal physical law, the foundation model applies different, seemingly nonsensical laws depending on the task it’s being applied to.
Notes
Check out this amazing precursor: “Se Simplicio avesse avuto un Cray” by Preparata (in Italian).
Mathematics: the Rise of the Machines¶
{2511.17203} Mathematics: the Rise of the Machines, Yang-Hui He (2025)
We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines in the mathematical sciences. At the heart is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.
Advisory note
◼ Dark side
Quotes
Even in a hypothetical distant future where complete mechanization is achieved, where AI will (dis-)prove every major conjecture and propose new problems and continuously map out new areas of research, human mathematicians are still of great value. We will simply become priests to oracles, and interpret the results to the rest of humanity. Think of the philosophy departments in the world, centuries are spent in analyzing and critiquing Plato, or the literature departments, over Shakespeare. Perhaps one day in the far future, mathematics departments will consist of experts digesting the (Mathlib-verified) proofs that AI produces. Still, I think a number of my colleagues agree with the exclamation [HLF] that “I don’t care whether it is God, AI, or Terry Tao who finds the proof of the Riemann Hypothesis, I just want to know.”
Notes
Who cares about whether the Riemann hypothesis is true or not? The point is why it is true or not. (Credit: Thomas Voigtmann).
Perfect score on IPhO 2025 theory by Gemini agent¶
{2603.03352} Perfect score on IPhO 2025 theory by Gemini agent, Yichen Huang (2026)
The International Physics Olympiad (IPhO) is the world’s most prestigious and renowned physics competition for pre-university students. IPhO problems require complex reasoning based on deep understanding of physical principles in a standard general physics curriculum. On IPhO 2025 theory problems, while gold medal performance by AI models was reported previously, it falls behind the best human contestant. Here we build a simple agent with Gemini 3.1 Pro Preview. We run it five times and it achieved a perfect score every time. However, data contamination could occur because Gemini 3.1 Pro Preview was released after the competition.
Quotes
Gemini 3.1 Pro Preview was released after IPhO 2025. Thus, IPhO 2025 problems could occur in the training dataset of the model, and the perfect performance by our Gemini agent should be interpreted with caution. However, it should be clear that our result is still meaningful. The second highest reported result 87.7% was achieved by Gemini 3 Deep Think [undefm]. In the release note [undefu], the Gemini team indicates that Gemini 3.1 Pro is the “upgraded core intelligence that makes [the Gemini 3 Deep Think] breakthroughs possible.” Thus, the risk of data contamination for Gemini 3 Deep Think is at the same level as that for our agent built on Gemini 3.1 Pro Preview.