AI for Science: January 2026 Week 5

1. Test-time Generalization for Physics through Neural Operator Splitting

arXiv:2602.00884

Why Novel: Achieves true zero-shot generalization for neural operators on unseen PDE dynamics without modifying pretrained weights, by introducing compositional operator splitting that searches over combinations of trained operators at test time.

Key Innovations:

• Formulates test-time generalization as compositional search over a DISCO dictionary of pretrained neural operators
• Introduces neural operator splitting strategy that decomposes unseen dynamics into compositions of known operators (e.g., $f_{OOD} \approx f_{i_1} + f_{i_2}$)
• Develops beam search and uniform search strategies to efficiently explore operator compositions
• Enables PDE parameter identification from test-time observations without retraining
• Achieves state-of-the-art zero-shot generalization: 0.015 NRMSE on advection-diffusion vs 0.170 for DISCO baseline

Evidence:

• tab:1 — Zero-shot generalization to unseen PDE combinations showing 10x improvement over baselines on multi-physics tasks
• tab:2 — Parameter extrapolation results: beam search achieves 0.002 NRMSE vs 0.159 for DISCO on diffusion coefficient
• fig:1 — Architecture overview showing operator dictionary construction and test-time composition search
• fig:3 — Test-time scaling laws showing continuous improvement with more search trials and accurate parameter recovery

Impact: Establishes test-time computation as a powerful paradigm for neural operator generalization, demonstrating that compositional structure enables flexible adaptation to unseen physics without expensive retraining.

2. Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks

arXiv:2601.19818

Why Novel: First framework providing mathematically rigorous, machine-verifiable error bounds for PINN solutions of ODEs, addressing the fundamental trustworthiness gap in neural network-based scientific computing.

Key Innovations:

• Introduces Doubly Smoothed Maximum (DSM) loss enabling learning of sub- and super-solutions that provably bound the true solution
• Combines neural network training with interval arithmetic verification for gap-free mathematical certification
• Provides computable a posteriori error bounds as machine-verifiable proofs without requiring closed-form references
• Extends to blow-up problems, computing rigorous bounds on finite-time blow-up times (e.g., Riccati equation)
• Achieves 98-100% verification success rates on logistic and generalized logistic equations with appropriate regularization

Evidence:

• fig:1 — Learn and Verify framework overview showing the two-phase approach: approximate solution learning followed by interval verification
• thm:3 — Main theorem establishing conditions for verified enclosure of true ODE solutions
• tab:1 — Verification success rates reaching 100% at 300 epochs for tolerances down to $2^{-6}$

Impact: Bridges the gap between neural network flexibility and mathematical rigor, establishing a foundation for trustworthy scientific machine learning with formal guarantees essential for safety-critical applications.

3. Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

arXiv:2601.22123

Why Novel: Enables stable molecular dynamics simulation at 10-18x larger timesteps than classical integrators by learning mean phase-space evolution directly from trajectory-free samples, eliminating the need for expensive reference trajectory generation.

Key Innovations:

• Formulates Hamiltonian Flow Maps predicting mean velocity $\bar{v}$ and mean force $\bar{f}$ over arbitrary time intervals
• Introduces Mean Flow consistency objective enabling training on decorrelated ab-initio samples without trajectory data
• Develops inference-time filters for drift removal, energy-momentum conservation, and approximate rotation equivariance
• Achieves comparable accuracy to standard MLFF at 0.5 fs while running at 9 fs timesteps on molecular systems
• Demonstrates 4x fewer integration steps needed to match Velocity Verlet accuracy on N-body systems

Evidence:

• tab:1 — Interatomic distances MAE showing HFM at 9 fs matches or approaches MLFF baseline at 0.5 fs
• fig:5 — N-body rollout showing HFM maintains accuracy at coarse discretization while VV diverges rapidly
• fig:1 — Architecture overview contrasting trajectory-based approaches with trajectory-free HFM training
• fig:6 — State space exploration showing HFM covers configuration space much faster due to larger timesteps

Impact: Provides a practical pathway to accelerating molecular dynamics simulations by an order of magnitude while maintaining data efficiency, with broad applicability to drug discovery, materials science, and computational chemistry.

1. Elign: Equivariant Diffusion Model Alignment from Foundational Machine Learning Force Fields

arXiv:2601.21985

Introduces RLHF-style post-training alignment for molecular diffusion models using MLFF rewards, achieving physically stable conformations via Force-Energy Disentangled GRPO while maintaining unguided inference speed.

2. Generalizable Equivariant Diffusion Models for Non-Abelian Lattice Gauge Theory

arXiv:2601.19552

Demonstrates gauge-equivariant diffusion models can accurately sample U(2) and SU(2) lattice gauge theories using MAALA, generalizing remarkably well to larger couplings and lattice sizes from single-ensemble training.

3. Forward and Inverse Mantle Convection with Neural Operators

arXiv:2601.23178

Applies Fourier Neural Operators to mantle convection, enabling both physics-informed forward Stokes operators and data-driven long-range convection operators for thermal state reconstruction in geodynamics.

4. A Dynamic Framework for Grid Adaptation in Kolmogorov-Arnold Networks

arXiv:2601.18672

Proposes curvature-based Importance Density Functions for KAN grid adaptation, achieving 25% error reduction on synthetic functions and 23% on PDE benchmarks over input-density baselines.

5. Neural Theorem Proving for Verification Conditions: A Real-World Benchmark

arXiv:2601.18944

Introduces RealVC benchmark of 637 program verification conditions in Why3, exposing that state-of-the-art neural theorem provers struggle with SMT-arithmetic and quantifier handling in real-world VCs.

6. Quantum Random Features: A Spectral Framework for Quantum Machine Learning

arXiv:2601.21746

Presents QRF and QDRF as lightweight quantum reservoir models achieving 89.3% accuracy on Fashion-MNIST with $O(\log N_f)$ preprocessing cost, providing hardware-compatible QML without variational optimization.

7. Towards Agentic Intelligence for Materials Science

arXiv:2602.00169

Comprehensive survey positioning agentic LLM systems as the path to autonomous materials discovery, proposing frameworks for integrating reactive ML models into end-to-end discovery loops with human oversight.