Abstract
We develop a combined machine learning (ML) and quantum mechanics approach that enables data-efficient reconstruction of flexible molecular force fields from high-level ab initio calculations, through the consideration of fundamental physical constraints. We discuss how such constraints are recovered and incorporated into ML models. Specifically, we use conservation of energy—a fundamental property of closed classical and quantum mechanical systems—to derive an efficient gradient-domain machine learning (GDML) model. The challenge of constructing conservative force fields is accomplished by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. We proceed with the development of a multi-partite matching algorithm that enables a fully automated recovery of physically relevant point group and fluxional symmetries from the training dataset into a symmetric variant of our model. The symmetric GDML (sGDML) approach is able to faithfully reproduce global force fields at the accuracy high-level ab initio methods, thus enabling sample intensive tasks like molecular dynamics simulations at that level of accuracy. (This chapter is adapted with permission from Chmiela (Towards exact molecular dynamics simulations with invariant machine-learned models, PhD thesis. Technische Universität, Berlin, 2019).)
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Notes
- 1.
For illustrative purposes, we use the definition of curl in three dimensions here, but the theory directly generalizes to arbitrary dimension. One way to prove this is via path-independence of conservative vector fields: the circulation of a gradient along any closed curve is zero and the curl is the limit of such circulations.
References
M.E. Tuckerman, Ab initio molecular dynamics: basic concepts, current trends and novel applications. J. Phys. Condens. Matter 14(50), R1297 (2002)
M. Rupp, A. Tkatchenko, K.-R. Müller, O.A. Von Lilienfeld. Fast and accurate modeling of molecular atomization energies with machine learning. Phys. Rev. Lett. 108(5), 58301 (2012)
K. Hansen, G. Montavon, F. Biegler, S. Fazli, M. Rupp, M. Scheffler, O.A. von Lilienfeld, A. Tkatchenko, K.-R. Müller, Assessment and validation of machine learning methods for predicting molecular atomization energies. J. Chem. Theory Comput. 9(8), 3404–3419 (2013)
K. Hansen, F. Biegler, R. Ramakrishnan, W. Pronobis, O.A. von Lilienfeld, K.-R. Müller, A. Tkatchenko, Machine learning predictions of molecular properties: accurate many-body potentials and nonlocality in chemical space. J. Phys. Chem. Lett. 6(12), 2326–2331 (2015)
M. Rupp, R. Ramakrishnan, O.A. von Lilienfeld, Machine learning for quantum mechanical properties of atoms in molecules. J. Phys. Chem. Lett. 6(16), 3309–3313 (2015)
V. Botu, R. Ramprasad, Adaptive machine learning framework to accelerate ab initio molecular dynamics. Int. J. Quantum Chem. 115(16), 1074–1083 (2015)
M. Hirn, N. Poilvert, S. Mallat, Quantum energy regression using scattering transforms. CoRR, abs/1502.02077 (2015)
R. Ramakrishnan, P.O. Dral, M. Rupp, O.A. von Lilienfeld, Big data meets quantum chemistry approximations: the δ-machine learning approach. J. Chem. Theory Comput. 11(5), 2087–2096 (2015)
S. De, A.P. Bartók, G. Csányi, M. Ceriotti, Comparing molecules and solids across structural and alchemical space. Phys. Chem. Chem. Phys. 18(20), 13754–13769 (2016)
N. Artrith, A. Urban, G. Ceder, Efficient and accurate machine-learning interpolation of atomic energies in compositions with many species. Phys. Rev. B 96(1), 14112 (2017)
A.P. Bartók, S. De, C. Poelking, N. Bernstein, J.R. Kermode, G. Csányi, M. Ceriotti, Machine learning unifies the modeling of materials and molecules. Sci. Adv. 3(12), e1701816 (2017)
A. Glielmo, P. Sollich, A. De Vita, Accurate interatomic force fields via machine learning with covariant kernels. Phys. Rev. B 95, 214302 (2017)
K. Yao, J.E. Herr, J. Parkhill, The many-body expansion combined with neural networks. J. Chem. Phys. 146(1), 14106 (2017)
S.T. John, G. Csányi, Many-body coarse-grained interactions using Gaussian approximation potentials. J. Phys. Chem. B 121(48), 10934–10949 (2017)
F.A. Faber, L. Hutchison, B. Huang, J. Gilmer, S.S. Schoenholz, G.E. Dahl, O. Vinyals, S. Kearnes, P.F. Riley, O.A. von Lilienfeld, Prediction errors of molecular machine learning models lower than hybrid DFT error. J. Chem. Theory Comput. 13(11), 5255–5264 (2017)
M. Eickenberg, G. Exarchakis, M. Hirn, S. Mallat, L. Thiry, Solid harmonic wavelet scattering for predictions of molecule properties. J. Chem. Phys. 148(24), 241732 (2018)
A. Glielmo, C. Zeni, A. De Vita, Efficient nonparametric n-body force fields from machine learning. Phys. Rev. B 97(18), 184307 (2018)
Y.-H. Tang, D. Zhang, G. Em Karniadakis, An atomistic fingerprint algorithm for learning ab initio molecular force fields. J. Chem. Phys. 148(3), 34101 (2018)
A. Grisafi, D.M. Wilkins, G. Csányi, M. Ceriotti, Symmetry-adapted machine learning for tensorial properties of atomistic systems. Phys. Rev. Lett. 120, 36002 (2018)
W. Pronobis, A. Tkatchenko, K.-R. Müller, Many-body descriptors for predicting molecular properties with machine learning: analysis of pairwise and three-body interactions in molecules. J. Chem. Theory Comput. 14(6), 2991–3003 (2018)
J. Behler, M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98(14), 146401 (2007)
A.P. Bartók, M.C. Payne, R. Kondor, G. Csányi, Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. Phys. Rev. Lett. 104(13), 136403 (2010)
K.V. Jovan Jose, N. Artrith, J. Behler, Construction of high-dimensional neural network potentials using environment-dependent atom pairs. J. Chem. Phys. 136(19), 194111 (2012)
A.P. Bartók, R. Kondor, G. Csányi, On representing chemical environments. Phys. Rev. B 87(18), 184115 (2013)
G. Montavon, M. Rupp, V. Gobre, A. Vazquez-Mayagoitia, K. Hansen, A. Tkatchenko, K.-R. Müller, O.A. von Lilienfeld, Machine learning of molecular electronic properties in chemical compound space. New J. Phys. 15(9), 95003 (2013)
A.P. Bartók, G. Csányi, Gaussian approximation potentials: a brief tutorial introduction. Int. J. Quantum Chem. 115(16), 1051–1057 (2015)
V. Botu, R. Ramprasad, Learning scheme to predict atomic forces and accelerate materials simulations. Phys. Rev. B 92, 94306 (2015)
T. Bereau, D. Andrienko, O.A. von Lilienfeld, Transferable atomic multipole machine learning models for small organic molecules. J. Chem. Theory Comput. 11(7), 3225–3233 (2015)
Z. Li, J.R. Kermode, A. De Vita, Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces. Phys. Rev. Lett. 114, 96405 (2015)
J. Behler, Perspective: machine learning potentials for atomistic simulations. J. Chem. Phys. 145(17), 170901 (2016)
F. Brockherde, L. Vogt, L. Li, M.E. Tuckerman, K. Burke, K.-R. Müller, Bypassing the Kohn-Sham equations with machine learning. Nat. Commun. 8, 872 (2017)
M. Gastegger, J. Behler, P. Marquetand, Machine learning molecular dynamics for the simulation of infrared spectra. Chem. Sci. 8, 6924–6935 (2017)
K.T. Schütt, F. Arbabzadah, S. Chmiela, K.-R. Müller, A. Tkatchenko, Quantum-chemical insights from deep tensor neural networks. Nat. Commun. 8, 13890 (2017)
K. Schütt, P.-J. Kindermans, H.E. Sauceda, S. Chmiela, A. Tkatchenko, K.-R. Müller, SchNet: a continuous-filter convolutional neural network for modeling quantum interactions, in Advances in Neural Information Processing Systems, vol. 31, pp. 991–1001 (2017)
K.T. Schütt, H.E. Sauceda, P.-J. Kindermans, A. Tkatchenko, K.-R. Müller, SchNet—A deep learning architecture for molecules and materials. J. Chem. Phys. 148(24), 241722 (2018)
B. Huang, O.A. von Lilienfeld, The “DNA” of chemistry: scalable quantum machine learning with “amons”. arXiv preprint:1707.04146 (2017)
T.D. Huan, R. Batra, J. Chapman, S. Krishnan, L. Chen, R. Ramprasad, A universal strategy for the creation of machine learning-based atomistic force fields. NPJ Comput. Mater. 3(1), 37 (2017)
E.V. Podryabinkin, A.V. Shapeev, Active learning of linearly parametrized interatomic potentials. Comput. Mater. Sci. 140, 171–180 (2017)
P.O. Dral, A. Owens, S.N. Yurchenko, W. Thiel, Structure-based sampling and self-correcting machine learning for accurate calculations of potential energy surfaces and vibrational levels. J. Chem. Phys. 146(24), 244108 (2017)
L. Zhang, J. Han, H. Wang, R. Car, E. Weinan, Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120(14), 143001 (2018)
N. Lubbers, J.S. Smith, K. Barros, Hierarchical modeling of molecular energies using a deep neural network. J. Chem. Phys. 148(24), 241715 (2018)
K. Ryczko, K. Mills, I. Luchak, C. Homenick, I. Tamblyn, Convolutional neural networks for atomistic systems. Comput. Mater. Sci. 149, 134–142 (2018)
K. Kanamori, K. Toyoura, J. Honda, K. Hattori, A. Seko, M. Karasuyama, K. Shitara, M. Shiga, A. Kuwabara, I. Takeuchi, Exploring a potential energy surface by machine learning for characterizing atomic transport. Phys. Rev. B 97(12), 125124 (2018)
T.S. Hy, S. Trivedi, H. Pan, B.M. Anderson, R. Kondor, Predicting molecular properties with covariant compositional networks. J. Chem. Phys. 148(24), 241745 (2018)
J. Wang, S. Olsson, C. Wehmeyer, A. Pérez, N.E. Charron, G. De Fabritiis, F. Noé, C. Clementi, Machine learning of coarse-grained molecular dynamics force fields. ACS Cent. Sci. 5(5), 755–767 (2019)
T. Bereau, R.A. DiStasio Jr., A. Tkatchenko, O.A. Von Lilienfeld, Non-covalent interactions across organic and biological subsets of chemical space: physics-based potentials parametrized from machine learning. J. Chem. Phys. 148(24), 241706 (2018)
A. Mardt, L. Pasquali, H. Wu, F. Noé, VAMPnets for deep learning of molecular kinetics. Nat. Commun. 9(1), 5 (2018)
F. Noé, S. Olsson, J. Köhler, H. Wu, Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science 365(6457), eaaw1147 (2019)
N. Thomas, T. Smidt, S. Kearnes, L. Yang, L. Li, K. Kohlhoff, P. Riley, Tensor field networks: rotation-and translation-equivariant neural networks for 3D point clouds. arXiv preprint:1802.08219 (2018)
J.S. Smith, B. Nebgen, N. Lubbers, O. Isayev, A. Roitberg, Less is more: sampling chemical space with active learning. J. Chem. Phys. 148(24), 241733 (2018)
K. Gubaev, E.V. Podryabinkin, A.V. Shapeev, Machine learning of molecular properties: locality and active learning. J. Chem. Phys. 148(24), 241727 (2018)
F.A. Faber, A.S. Christensen, B. Huang, O.A. von Lilienfeld, Alchemical and structural distribution based representation for universal quantum machine learning. J. Chem. Phys. 148(24), 241717 (2018)
A.S. Christensen, F.A. Faber, O.A. von Lilienfeld, Operators in quantum machine learning: response properties in chemical space. J. Phys. Chem. 150(6), 64105 (2019)
R. Winter, F. Montanari, F. Noé, D.-A. Clevert, Learning continuous and data-driven molecular descriptors by translating equivalent chemical representations. Chem. Sci. 10(6), 1692–1701 (2019)
K. Gubaev, E.V. Podryabinkin, G.L.W. Hart, A.V. Shapeev, Accelerating high-throughput searches for new alloys with active learning of interatomic potentials. Comput. Mater. Sci. 156, 148–156 (2019)
E. Noether, Invarianten beliebiger Differentialausdrücke. Gött. Nachr. Mathematisch-Physikalische Klasse 1918, 37–44 (1918)
K.-R. Müller, S. Mika, G. Rätsch, K. Tsuda, B. Schölkopf, An introduction to kernel-based learning algorithms. IEEE Trans. Neural Netw. Learn. Syst. 12(2), 181–201 (2001)
B. Schölkopf, A.J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (MIT Press, Cambridge, 2002)
G. Wahba, Spline Models for Observational Data, vol. 59 (SIAM, Philadelphia, 1990)
B. Schölkopf, R. Herbrich, A.J. Smola, A generalized representer theorem, in International Conference on Computational Learning Theory (Springer, Berlin, 2001), pp. 416–426
C.E. Rasmussen, Gaussian processes in machine learning, in Advanced Lectures on Machine Learning (Springer, Berlin, 2004), pp. 63–71
D. Duvenaud, Automatic Model Construction with Gaussian Processes, PhD thesis, University of Cambridge, Cambridge, 2014
C.A. Micchelli, Y. Xu, H. Zhang, Universal kernels. J. Mach. Learn. Res. 7(Dec), 2651–2667 (2006)
A. Damianou, N. Lawrence, Deep Gaussian processes, in Artificial Intelligence and Statistics (2013), pp. 207–215
C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1949)
K. Brading, E. Castellani, Symmetries in Physics: Philosophical Reflections (Cambridge University Press, Cambridge, 2003)
D.J.C. MacKay, Introduction to Gaussian processes, in NATO ASI Series F: Computer and Systems Sciences, vol. 168 (Springer, Berlin, 1998)
A.J. Smola, B. Schölkopf, K.-R. Müller, The connection between regularization operators and support vector kernels. Neural Netw. 11(4), 637–649 (1998)
C. Heil, Metrics, Norms, Inner Products, and Operator Theory (Birkhäuser, Basel, 2018)
A. Rahimi, B. Recht, Random features for large-scale kernel machines, in Advances in Neural Information Processing Systems (2008), pp. 1177–1184
P. Politzer, J.S. Murray, The Hellmann-Feynman theorem: a perspective. J. Mol. Model. 24(9), 266 (2018)
R.P. Feynman, Forces in molecules. Phys. Rev. 56(4), 340 (1939)
C.E. Shannon, Communication in the presence of noise. Proc. IEEE 86(2), 447–457 (1998)
S. Chmiela, Towards Exact Molecular Dynamics Simulations with Invariant Machine-Learned Models, PhD thesis. Technische Universität, Berlin, 2019
T. Hastie, R. Tibshirani, J.H. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics (Springer, Berlin, 2009)
M.A. Alvarez, L. Rosasco, N.D. Lawrence, et al., Kernels for vector-valued functions: a review. Found. Trends Mach. Learn. 4(3), 195–266 (2012)
P. Boyle, M. Frean, Dependent Gaussian processes, in Advances in Neural Information Processing Systems (2005), pp. 217–224
C.A. Micchelli, M. Pontil, On learning vector-valued functions. Neural Comput. 17(1), 177–204 (2005)
C.A. Micchelli, M. Pontil, Kernels for multi-task learning, in Advances in Neural Information Processing Systems (2005), pp. 921–928
L. Baldassarre, L. Rosasco, A. Barla, A. Verri, Multi-output learning via spectral filtering. Mach. Learn. 87(3), 259–301 (2012)
T. Graepel, Solving noisy linear operator equations by Gaussian processes: application to ordinary and partial differential equations, in International Conference on Machine Learning (2003), pp. 234–241
S. Särkkä, Linear operators and stochastic partial differential equations in Gaussian process regression, in International Conference on Artificial Neural Networks (Springer, Berlin, 2011), pp. 151–158
E.M. Constantinescu, M. Anitescu, Physics-based covariance models for Gaussian processes with multiple outputs. Int. J. Uncertain. Quantif. 3(1) (2013)
N.C. Nguyen, J. Peraire, Gaussian functional regression for linear partial differential equations. Comput. Methods Appl. Mech. Eng. 287, 69–89 (2015)
C. Jidling, N. Wahlström, A. Wills, T.B. Schön, Linearly constrained Gaussian processes, in Advances in Neural Information Processing Systems (2017), pp. 1215–1224
F.J. Narcowich, J.D. Ward, Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comput. 63(208), 661–687 (1994)
E. Solak, R. Murray-Smith, W.E. Leithead, D.J. Leith, C.E. Rasmussen, Derivative observations in Gaussian process models of dynamic systems, in Advances in Neural Information Processing Systems (2003), pp. 1057–1064
S. Chmiela, A. Tkatchenko, H.E. Sauceda, I. Poltavsky, K.T. Schütt, K.-R. Müller, Machine learning of accurate energy-conserving molecular force fields. Sci. Adv. 3(5), e1603015 (2017)
S. Chmiela, H.E. Sauceda, I. Poltavsky, K.-R. Müller, A. Tkatchenko, sGDML: constructing accurate and data efficient molecular force fields using machine learning. Comput. Phys. Commun. 240, 38–45 (2019)
H.C. Longuet-Higgins, The symmetry groups of non-rigid molecules. Mol. Phys. 6(5), 445–460 (1963)
E.B. Wilson, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra (McGraw-Hill Interamericana, New York, 1955)
D. Pachauri, R. Kondor, V. Singh, Solving the multi-way matching problem by permutation synchronization, in Advances in Neural Information Processing Systems (2013), pp. 1860–1868
M. Schiavinato, A. Gasparetto, A. Torsello, Transitive Assignment Kernels for Structural Classification (Springer, Cham, 2015), pp. 146–159
N.M. Kriege, P.-L. Giscard, R.C. Wilson, On valid optimal assignment kernels and applications to graph classification, in Advances in Neural Information Processing Systems, vol. 30 (2016), pp. 1623–1631
J.-P. Vert, The optimal assignment kernel is not positive definite. CoRR, abs/0801.4061 (2008)
S. Umeyama, An eigendecomposition approach to weighted graph matching problems. IEEE Trans. Pattern Anal. Mach. Intell. 10(5), 695–703 (1988)
H.W. Kuhn, The Hungarian method for the assignment problem. Nav. Res. Logist. 2(1–2), 83–97 (1955)
L. van der Maaten, G. Hinton, Visualizing data using t-SNE. J. Mach. Learn. Res. 9(2579–2605), 85 (2008)
T. Karvonen, S. Särkkä, Fully symmetric kernel quadrature. SIAM J. Sci. Comput. 40(2), A697–A720 (2018)
B. Haasdonk, H. Burkhardt, Invariant kernel functions for pattern analysis and machine learning. Mach. Learn. 68(1), 35–61 (2007)
S. Chmiela, H.E. Sauceda, K.-R. Müller, A. Tkatchenko, Towards exact molecular dynamics simulations with machine-learned force fields. Nat. Commun. 9(1), 3887 (2018)
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Chmiela, S., Sauceda, H.E., Tkatchenko, A., Müller, KR. (2020). Accurate Molecular Dynamics Enabled by Efficient Physically Constrained Machine Learning Approaches. In: Schütt, K., Chmiela, S., von Lilienfeld, O., Tkatchenko, A., Tsuda, K., Müller, KR. (eds) Machine Learning Meets Quantum Physics. Lecture Notes in Physics, vol 968. Springer, Cham. https://doi.org/10.1007/978-3-030-40245-7_7
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