analytical second order derivatives (Hessians) - #15
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…second order derivatives of the potential energy (Hessians) in addition to energies and gradients. `test_torchtools_hessian.py` demonstrates that the new implementation of GDMLPredict(...) gives the same energies and gradients as the original GDMLTorchPredict(...) . The analytical Hessians are compared to numerical Hessian computed with ASE. Certain semiclassical propagators such as the Herman-Kluk propagator require a local harmonic approximation around each trajectory. Computing the Hessian numerically for each time step would be too inaccurate and time-consuming. The analytical Hessian calculation is roughly 5-10 times more expensive than a gradient calculation.
… the energy label is read as the first float in the line
…ordinary xyz file
…l index arrays to the GPU a 10x speed-up is achieved
…l index arrays to the GPU a 10x speed-up is achieved
… more expensive than a gradient calculation
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Semiclassical propagators such as the Herman-Kluk propagator (https://doi.org/10.1063/1.450142) require a local
harmonic approximation to the potential energy surface around the instantaneous position of each trajectory. Computing the Hessian numerically for each time step during a molecular dynamics simulation would be too inaccurate and time-consuming. Being able to compute Hessians efficiently is probably also useful for other applications such as optimizations of the molecular geometry.
torchtools_hessian.pycontains another torch implementation of the sGDML predictor, which in addition to energies and gradients/forces can also evaluate the Hessians analytically for a batch of geometries. This is done by differentiating the kernel a third time w/r/t to the cartesian coordinates. It is assumed that the descriptor is the Coulomb matrix.An analytical Hessian calculation is roughly 4 times more expensive than a gradient calculation.
test_torchtools_hessian.pychecks that the proposed predictor produces the same energies and gradients as the current implementation. It is a little bit slower than the current implementation when only energies and gradients are requested. The analytical Hessians are checked against finite-difference Hessians computed with ASE.I hope this might be useful.