Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics
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75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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Transforming Legacy Equation of State Databases into Interatomic Potentials via Machine Learning
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Salas, Ember Sikorski [Sandia National Laboratorie↗
Exact enforcement of temporal continuity in sequential physics-informed neural networks
The use of deep learning methods in scientific computing represents a potential paradigm shift in engineering problem solving. One of the most prominent developments is Physics-Informed Neural Networks (PINNs), in which neural networks are trained to satisfy partial differential equations (PDEs). While this method shows promise, the standard version has been shown to struggle in accurately predicting the dynamic behavior of time-dependent problems. To address this challenge, methods have been proposed that decompose the time domain into multiple segments, employing a distinct neural network in each segment and directly incorporating continuity between them in the loss function of the minimization problem. In this work we introduce a method to exactly enforce continuity between successive time segments via a solution ansatz. This hard constrained sequential PINN (HCS-PINN) method is simple to implement and eliminates the need for any loss terms associated with temporal continuity. The method is tested for a number of benchmark problems involving both linear and non-linear PDEs. Examples include various first order time dependent problems in which traditional PINNs struggle, namely advection, Allen–Cahn, and Korteweg–de Vries equations. Furthermore, second and third order time-dependent problems are demonstrated via wave and Jerky dynamics examples, respectively. Notably, the Jerky dynamics problem is chaotic, making the problem especially sensitive to temporal accuracy. Finally, the numerical experiments conducted with the proposed method demonstrated superior convergence and accuracy over both traditional PINNs and the soft-constrained counterparts.
42 ENGINEERING↗
Adjoint-operators and non-adiabatic learning algorithms in neural networks
Adjoint sensitivity equations are presented, which can be solved simultaneously (i.e., forward in time) with the dynamics of a nonlinear neural network. These equations provide the foundations for a new methodology which enables the implementation of temporal learning algorithms in a highly efficient manner.
Toomarian, N.↗
Machine learning approaches for structural and thermodynamic properties of a Lennard-Jones fluid
Predicting the functional properties of many molecular systems relies on understanding how atomistic interactions give rise to macroscale observables. However, current attempts to develop predictive models for the structural and thermodynamic properties of condensed-phase systems often rely on extensive parameter fitting to empirically selected functional forms whose effectiveness is limited to a narrow range of physical conditions. Here, we illustrate how these traditional fitting paradigms can be superseded using machine learning. Specifically, we use the results of molecular dynamics simulations to train machine learning protocols that are able to produce the radial distribution function, pressure, and internal energy of a Lennard-Jones fluid with increased accuracy in comparison to previous theoretical methods. The radial distribution function is determined using a variant of the segmented linear regression with the multivariate function decomposition approach developed by Craven et al. [J. Phys. Chem. Lett. 11, 4372 (2020)]. The pressure and internal energy are determined using expressions containing the learned radial distribution function and also a kernel ridge regression process that is trained directly on thermodynamic properties measured in simulation. The presented results suggest that the structural and thermodynamic properties of fluids may be determined more accurately through machine learning than through human-guided functional forms.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems
Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Lastly, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.
97 MATHEMATICS AND COMPUTING↗
Embedding hard physical constraints in neural network coarse-graining of three-dimensional turbulence
In recent years, deep learning approaches have shown much promise in modeling complex systems in the physical sciences. A major challenge in deep learning of partial differential equations is enforcing physical constraints and boundary conditions. In this work, we propose a general framework to directly embed the notion of an incompressible fluid into convolutional neural networks, and apply this to coarse-graining of turbulent flow. These physics-embedded neural networks leverage interpretable strategies from numerical methods and computational fluid dynamics to enforce physical laws and boundary conditions by taking advantage the mathematical properties of the underlying equations. Here, we demonstrate results on three-dimensional fully developed turbulence, showing that this technique drastically improves local conservation of mass, without sacrificing performance according to several other metrics characterizing the fluid flow.
97 MATHEMATICS AND COMPUTING↗
On fast simulation of dynamical system with neural vector enhanced numerical solver
The large-scale simulation of dynamical systems is critical in numerous scientific and engineering disciplines. However, traditional numerical solvers are limited by the choice of step sizes when estimating integration, resulting in a trade-off between accuracy and computational efficiency. To address this challenge, we introduce a deep learning-based corrector called Neural Vector (NeurVec), which can compensate for integration errors and enable larger time step sizes in simulations. Our extensive experiments on a variety of complex dynamical system benchmarks demonstrate that NeurVec exhibits remarkable generalization capability on a continuous phase space, even when trained using limited and discrete data. NeurVec significantly accelerates traditional solvers, achieving speeds tens to hundreds of times faster while maintaining high levels of accuracy and stability. Moreover, NeurVec’s simple-yet-effective design, combined with its ease of implementation, has the potential to establish a new paradigm for fast-solving differential equations based on deep learning.
97 MATHEMATICS AND COMPUTING↗
Scalable learning of potentials to predict time-dependent Hartree–Fock dynamics
We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.
97 MATHEMATICS AND COMPUTING↗
Bayesian operator inference for data-driven reduced-order modeling
This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.
97 MATHEMATICS AND COMPUTING↗
Accelerating multilevel Markov Chain Monte Carlo using machine learning models
Here, this work presents an efficient approach for accelerating multilevel Markov Chain Monte Carlo (MCMC) sampling for large-scale problems using low-fidelity machine learning models. While conventional techniques for large-scale Bayesian inference often substitute computationally expensive high-fidelity models with machine learning models, thereby introducing approximation errors, our approach offers a computationally efficient alternative by augmenting high-fidelity models with low-fidelity ones within a hierarchical framework. The multilevel approach utilizes the low-fidelity machine learning model (MLM) for inexpensive evaluation of proposed samples thereby improving the acceptance of samples by the high-fidelity model. The hierarchy in our multilevel algorithm is derived from geometric multigrid hierarchy. We utilize an MLM to accelerate the coarse level sampling. Training machine learning model for the coarsest level significantly reduces the computational cost associated with generating training data and training the model. We present an MCMC algorithm to accelerate the coarsest level sampling using MLM and account for the approximation error introduced. We provide theoretical proofs of detailed balance and demonstrate that our multilevel approach constitutes a consistent MCMC algorithm. Additionally, we derive the expression for cost reduction due to machine learning model to facilitate cost analysis of the hierarchical sampling algorithm. Our technique is demonstrated on a standard benchmark inference problem in groundwater flow, where we estimate the probability density of a quantity of interest using a four-level MCMC algorithm. Our proposed algorithm accelerates multilevel sampling by a factor of two while achieving similar accuracy compared to sampling using the standard multilevel algorithm.
97 MATHEMATICS AND COMPUTING↗
Unstructured-grid methods development: Lessons le arned
The development is summarized of unstructured grid methods for the solution of the equations of fluid flow and some of the lessons learned are shared. The 3-D Euler equations are solved, including spatial discretizations, temporal discretizations, and boundary conditions. An example calculation with an upwind implicit method using a CFL (Courant Friedricks Lewy) number of infinity is presented for the Boeing 747 aircraft. The results obtained in less than one hour of CPU time on a Cray-2 computer, thus demonstrating the speed and robustness of the present capability.
Batina, John T.↗
Machine-learning force-field models for dynamical simulations of metallic magnets
We review recent advances in machine-learning (ML) force-field methods for Landau–Lifshitz–Gilbert simulations of itinerant electron magnets, focusing on their scalability and transferability. Built on the principle of locality, a deep neural-network model is developed to efficiently and accurately predict electron-mediated forces governing spin dynamics. Symmetry-aware descriptors constructed through a group-theoretical approach ensure rigorous incorporation of both lattice and spin-rotation symmetries. The framework is demonstrated using the prototypical s-d exchange model widely employed in spintronics. ML-enabled large-scale simulations reveal novel nonequilibrium phenomena, including anomalous coarsening of tetrahedral spin order on the triangular lattice and the freezing of phase-separation dynamics in lightly hole-doped, strong-coupling square-lattice systems. These results establish ML force-field frameworks as scalable, accurate, and versatile tools for modeling nonequilibrium spin dynamics in itinerant magnets.
Artificial neural networks↗
Scaling Field-Theoretic Simulation for Multicomponent Mixtures with Neural Operators
Multicomponent polymer mixtures are ubiquitous in biological self-organization but are notoriously difficult to study computationally. Plagued by both slow single molecule relaxation times and slow equilibration within dense mixtures, molecular dynamics simulations are typically infeasible at the spatial scales required to study the stability of mesophase structure. Polymer field theories offer an attractive alternative, but analytical calculations are only tractable for mean-field theories and nearby perturbations, constraints that become especially problematic for fluctuation-induced effects such as coacervation. Here, we show that a recently developed technique for obtaining numerical solutions to partial differential equations based on operator learning, neural operators, lends itself to a highly scalable training strategy by parallelizing per-species operator maps. We illustrate the efficacy of our approach on six-component mixtures with randomly selected compositions and that it significantly outperforms the state-of-the-art pseudospectral integrators for field-theoretic simulations, especially as polymer lengths become long.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A scalable framework for learning the geometry-dependent solution operators of partial differential equations
Not Available
Yin, Minglang (ORCID:000000017861350X)↗
Solving Newton’s equations of motion with large timesteps using recurrent neural networks based operators
Classical molecular dynamics simulations are based on solving Newton’s equations of motion. Using a small timestep, numerical integrators such as Verlet generate trajectories of particles as solutions to Newton’s equations. We introduce operators derived using recurrent neural networks that accurately solve Newton’s equations utilizing sequences of past trajectory data, and produce energy-conserving dynamics of particles using timesteps up to 4000 times larger compared to the Verlet timestep. We demonstrate significant speedup in many example problems including 3D systems of up to 16 particles.
Newton’s equations↗
Predicting initial dissolution rates using structural features from molecular dynamics simulations
Predicting chemical durability of glass materials is important for various applications from daily life such as drink glass and kitchen ware to advanced technologies such as nuclear waste disposal and biomedicine. In this work, we explored prediction of initial dissolution rate through structural features from molecular dynamics (MD) simulations for a wide range of glass compositions (total 28) including borosilicate and aluminosilicate glasses, ZrO 2 -containing and V 2 O 5 -containing boroaluminosilicate glasses. The initial dissolution rates (r 0 ) measured experimentally at 90 °C with varying solution conditions were correlated with structural features (e.g., polyhedral linkages and non-bridging oxygen species) obtained from MD simulations, either from this study or from literature. Since hydrolysis of the glass network through breaking of the network former linkages (e.g., Si-O-Si, Si-O-Al, etc.) is a critical step of network glass dissolution, the statistics of these linkages obtained from MD were correlated to r0 through linear regression, where the coefficient of determination (R 2 ) and root mean square error are found to be 0.949 and 0.681, respectively. This model was compared and discussed with existing models developed by various approaches including machine learning, the kinetic rate equation, topological constraint theory, and other descriptors from MD simulations. The discussion provides insights on future model improvements to predict glass dissolution. In addition, the impact of V 2 O 5 on the glass dissolution was examined in detail, implicating that the impact is not the same across all glass compositions and test conditions.
36 MATERIALS SCIENCE↗