AI in Physical Sciences and Mathematics

Climate Sciences and Meteorology

GraphCast (DeepMind, 2023)

• Science: "Learning skillful medium-range global weather forecasting" (2023)
• Architecture: GNN (Graph Neural Network) on an icosahedral mesh
• Data: ERA5 (40 years of meteorological data from ECMWF)
• Result: Beats ECMWF HRES (the world's most accurate operational model) on 90% of metrics
• Speed: 10-day forecast in 60 seconds (vs hours for physical models)
• Access: Open-source; operational at ECMWF since 2023

PanGu-Weather (Huawei, 2023)

• Nature: 2023
• Architecture: 3D Transformer (Earth-Specific Transformer)
• Result: Beats ECMWF in tropical cyclone forecasting; 24h forecast

FourCastNet (NVIDIA, 2022)

• arXiv: 2202.11214
• Architecture: Adaptive Fourier Neural Operators (AFNO)
• Speed: 45,000× faster than physical simulation with similar quality

AIFS (ECMWF, 2024)

• Authors: ECMWF itself
• Integration: ECMWF's first ML model in operation
• Data: ERA5 + its own reanalysis data

Aurora (Microsoft, 2024)

• arXiv: 2405.13063
• Generalist: A single model for multiple climate tasks
• Result: Beats ECMWF, GraphCast, PanGu on many metrics

---

Computational Physics

DeepMind AlphaFold → FermiNet / PauliNet

FermiNet (DeepMind, 2019)

• arXiv: 1909.02487
• Mechanism: Neural network to solve the many-electron Schrödinger equation
• Result: Ab-initio calculation of molecular energy more accurate than conventional methods

Neural Network Potentials (NNPs)

• Use: Replace DFT calculations in molecular simulations
• Frameworks: MACE, NequIP, PaiNN, SchNet
• Acceleration: 100–1000× over DFT; enables protein simulations in ns/day

---

Formal Mathematics

AlphaProof (DeepMind, 2024)

• Event: International Mathematical Olympiad (IMO) 2024
• Result: 4/6 problems solved formally (silver medal level)
• Mechanism: RL over Lean 4 (formal theorem prover) + AlphaGeometry
• Significance: First AI to solve IMO problems at medal level

AlphaGeometry (DeepMind, 2024)

• Nature: "Solving olympiad geometry without human demonstrations" (2024)
• Result: Solves 25/30 olympiad geometry problems (vs 25.9 for gold medal)
• Mechanism: Symbolic + LLM to generate auxiliary constructions

DeepSeek-Prover-V1.5 (DeepSeek, 2024)

• arXiv: 2408.08152
• Mechanism: Fine-tuning on Lean 4 proofs + RLVR + Monte Carlo Tree Search
• Result: 63.5% on MiniF2F (formal mathematics benchmark)

---

Formal Theorem Provers

Lean 4 (Microsoft Research / Lean Community)

• URL: lean-lang.org
• Use in AI: AlphaProof, DeepSeek-Prover train on Lean 4
• Mathlib4: 100K+ formalized theorems; foundation for training

Coq

• Focus: Formal software verification
• Use: Critical systems; Linux kernel (parts)

Isabelle/HOL

• Focus: Abstract mathematics + hardware verification
• Use: Intel verifies chips with Isabelle

---

Olympiad Mathematics with LLMs

Qwen2.5-Math

• arXiv: 2409.12122
• Training: 1T mathematical tokens; TIR (Tool-Integrated Reasoning)
• Result: Open-source SOTA on MATH, AMC10/12, AIME

InternLM-Math

• Focus: Formal and informal mathematics; integration with Lean 4

---

Quantum Computing + AI

AlphaQubit (DeepMind, 2024)

• Nature: "Decoding quantum errors with machine learning" (2024)
• Mechanism: Neural network to decode quantum errors in real time
• Result: Superior to classical decoders (MWPM) with the same hardware

Quantum ML (QML)

• Status: Active research; quantum advantage not yet clearly demonstrated for ML
• Frameworks: PennyLane, Qiskit Machine Learning

---

Nuclear Fusion

DeepMind + EPFL: Plasma Control via RL (2022)

• Nature: "Magnetic control of tokamak plasmas through deep reinforcement learning" (2022)
• Result: RL controls the TCV tokamak's plasma in configurations never before achieved
• Impact: Acceleration of fusion reactor development (ITER, Commonwealth Fusion)

---

Astrophysics

ML for Cataloging Galaxies

• DeepPSF, Morpheus: CNN to segment astronomical images
• Use: LSST (Vera Rubin Observatory) uses ML to classify billions of objects

GW (Gravitational Waves) — LIGO

• ML in LIGO: CNN to detect and classify gravitational-wave signals in noisy data
• Speed: 2,000× faster than classical matched filtering

---

Summary Table