From Atoms to Buildings, the Math Breaks

No AI or machine learning method can bridge atomic-to-continuum scales with provable accuracy guarantees. Predicting macroscopic material properties from atomic-scale physics requires coupling simulations across ~10 orders of magnitude in length and time — from quantum mechanics (angstroms, femtoseconds) through molecular dynamics (nanometers, nanoseconds) to continuum mechanics (meters, seconds). AI surrogate models trained on fine-grained simulation data can approximate coarse-grained dynamics, but no method provides provable error bounds on this coarse-graining, and no framework guarantees that critical rare events (phase transitions, crack nucleation, defect migration) are preserved in the surrogate.

Multi-scale modeling is essential for materials design, drug discovery, climate modeling, and engineering simulation. The Materials Genome Initiative has invested $500+ million since 2011 to accelerate materials discovery, but multi-scale prediction remains the central bottleneck — designing a new alloy or polymer still requires decades of iterative experiment because simulation cannot reliably predict bulk properties from atomic composition. A reliable AI multi-scale framework would compress materials development timelines from decades to years, with implications across aerospace, energy, medicine, and manufacturing.

Machine learning interatomic potentials (MLIPs — GAP, NequIP, MACE) accurately reproduce ab initio energies and forces but only at the atomistic scale — they don't bridge to continuum. Coarse-grained molecular dynamics with ML force fields can run larger systems faster but loses information about rare events and fails when the coarse-graining scheme encounters conditions outside its training distribution. Physics-informed neural networks (PINNs) solve PDEs but don't learn the PDEs — they require the governing equations to be known, which is exactly what multi-scale modeling tries to discover. Graph neural networks for materials property prediction (CGCNN, MEGNet) correlate structure to properties but are purely data-driven with no physics-based error bounds. The fundamental challenge is that coarse-graining is an irreversible information-losing projection — reconstructing fine-grained behavior from coarse-grained representations is an ill-posed inverse problem.

Mathematical theory for provably accurate coarse-graining — determining what information is necessarily lost in scale transitions and bounding the resulting prediction error. Data-efficient methods for learning rare-event dynamics from molecular simulations without requiring prohibitively long trajectories. Hybrid frameworks that couple physics-based models at each scale with learned scale-bridging operators, validated against experiment at each level.

A student team could take a well-studied system (e.g., water, a simple metal) where both atomistic and continuum behavior are well-characterized, train an ML surrogate model on molecular dynamics data, and systematically measure where the surrogate fails — identifying the types of phenomena that are lost in coarse-graining. Alternatively, a team could compare different ML interatomic potentials' predictions of a specific material property (e.g., thermal conductivity) against experimental measurements, quantifying the accuracy gap between atomic-scale ML and macroscopic observation. Relevant skills: computational materials science, machine learning, molecular dynamics, statistical mechanics.