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Colour me shocked: Exact Molecular Hessians from MLIPs in O(N) time using sparse differentiation!

NeurIPS 2026 · Architectures and efficiency › RNNs and normalization · approach: Scalable approximation and equivariant compute

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Shows that the Hessian sparsity pattern of local message-passing MLIPs can be derived in closed form from the cutoff graph and layer count, so sparse AD (star/acyclic colouring) yields exact Cartesian Hessians with a system-size-independent number of HVPs, giving O(N) total cost; packaged as ColPackPy.</tql>

Problem

Exact Cartesian Hessians of the energy require O(N) Hessian-vector products, so building MLIP Hessians costs O(N^2) overall, limiting high-accuracy Hessians to small systems. Sparse AD normally needs a costly sparsity-detection pass.

Approach

The receptive-field sparsity of local MLIPs (cutoff radius and number of message-passing layers) gives the Hessian sparsity pattern a priori; combining this with graph colouring (star or acyclic) reduces HVPs to a size-independent constant, so exact Hessians cost O(N) in total with no approximation. A heuristic cutoff-based pattern extends this to DFT and other non-local methods at the cost of app

Key results

• Acyclic colouring gives runtime speedups up to 15x on alkane chains and 8x on water clusters with one-layer TensorNet.
• Up to 6x runtime savings on Aβ40 conformers with one-layer TensorNet.
• With a cutoff-based approximate pattern, CHGNet and DFT (PBE) reach ~1e-6 to 1e-7 and ~0.1% average eigenvalue error respectively around a 17 Å cutoff on Chignolin.

Surprising findings

• Speedup is larger for MACE-OFF23 than TensorNet despite identical sparsity, because coloring overhead is a smaller share of the more expensive HVP cost.
• DFT Hessian error decreases more slowly with cutoff than CHGNet, suggesting longer-range dependencies in DFT Hessians.

Limitations

• Colouring runs on CPU and contributes a non-negligible share of runtime for cheap models.
• DFT and non-local models require heuristic cutoff patterns, introducing approximation error.
• Code is stated to be released only 'soon'; no link given.

Experimental setup

Datasets: Chignolin structures Aβ40 conformers from protein ensemble database

Benchmarks: alkane chains water clusters Aβ40 conformers Chignolin cutoff approximation

Metrics: HVP count wall-clock runtime average relative eigenvalue error maximum Hessian element error

Baselines: naive dense Hessian via HVPs star colouring

Builds on: ColPack sparse automatic differentiation asdex TensorNet CHGNet

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