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Deep learning and the Schrödinger equation

Kyle Mills, Michael Spanner, Isaac Tamblyn

arXiv:1702.01361v3cond-mat.mtrl-scics.LGphysics.chem-ph

TL;DR

The paper asks whether deep learning can predict electronic quantities directly from two-dimensional confining electrostatic potentials without hand-selected features. It trains a convolutional neural network on four potential classes using numerical ground truth, achieving chemical accuracy for random potentials while also evaluating kinetic and excited-state energies. The approach is promising within this confined-electron setting, but rotational invariance is not native to the architecture and efficient training and evaluation require substantial computational resources.

  • Problem

    Existing approaches often depend on hand-selected input representations, motivating a featureless method for learning electronic quantities directly from electrostatic potentials.

  • Method

    A convolutional deep neural network is trained on four classes of grid-represented confining two-dimensional potentials, using numerical ground-truth solutions.

  • Results

    1.49 mHa median absolute error for ground-state energy on random potentials, below the 1.6 mHa chemical-accuracy threshold.

  • Takeaways & Limitations

    The results support deep convolutional networks as promising candidates for rapidly approximating electronic-structure quantities and related partial-differential-equation solutions within the studied scope.

  • Takeaways & Limitations

    Efficient training and evaluation require substantial computational resources, and the convolutional network does not natively encode rotational invariance.

Abstract

from arXiv · show

We have trained a deep (convolutional) neural network to predict the ground-state energy of an electron in four classes of confining two-dimensional electrostatic potentials. On randomly generated potentials, for which there is no analytic form for either the potential or the ground-state energy, the neural network model was able to predict the ground-state energy to within chemical accuracy, with a median absolute error of 1.49 mHa. We also investigate the performance of the model in predicting other quantities such as the kinetic energy and the first excited-state energy of random potentials.

INTRODUCTION

The paper proposes featureless deep learning to learn mappings from electrostatic potentials to electronic quantities without hand-selected representations or explicit wavefunction computation. A convolutional neural network is demonstrated on confined-electron problems as a possible scalable approach to electronic structure calculations.

  • INTRODUCTION: Hand-selected features can limit transferability because performance depends on the input representation and whether all relevant features are included.
  • INTRODUCTION: Featureless learning trains a flexible deep model to learn both relevant features and the mapping from potential inputs to electronic outputs.The same data are supplied to the neural network and the numerical method, avoiding manual feature selection.
  • INTRODUCTION: The approach bypasses numerical solution of the Schrödinger equation and explicit computation of wavefunctions when predicting energy.
  • INTRODUCTION: The authors suggest that this approach may provide a more scalable and parallelizable route to large-scale electronic structure problems.
  • INTRODUCTION: The convolutional network learns mappings from confining electrostatic potentials to ground-state energy, kinetic energy, and first excited-state energy.

Training set: choice of potentials

The training data span four classes of two-dimensional confining electrostatic potentials, represented as grayscale-like grids of floating-point values. The dataset choice is presented as central to the model’s transferability.

  • Training set: choice of potentials: The study investigates simple harmonic oscillators, infinite wells, double-well inverted Gaussians, and random potentials.Each potential is represented as a grid of floating-point numbers, analogous to a grayscale image.

Numerical solver

A finite-difference solver provides ground-truth eigenvalues for potentials ranging from analytically tractable examples to random cases without closed-form potentials or eigenenergies. The random set is intended as the most demanding test of learned potential-to-energy mappings.

  • Numerical solver: A finite-difference method is used to solve the eigenvalue problem and generate reference values for training and evaluation.
  • Numerical solver: Figure 2 shows probability densities |ψ0|2 alongside their corresponding potentials for two random-potential examples.
  • Numerical solver: The potentials are discretized on a 256 × 256 grid over a square domain from −20 to 20 a.u., with atomic units used throughout.For simple harmonic oscillators, the calculated energies differ from analytic references by a median absolute error of 0.12 mHa.
  • Numerical solver: Simple harmonic oscillator energies depend on the two curvature parameters kx and ky, whereas infinite-well energies depend on the well widths in the two dimensions.These mappings are comparatively simple for a neural network to discover.
  • Numerical solver: Double-well inverted Gaussian potentials are more complex because several Gaussian parameters affect the energy and no analytic single-electron solution is known.
  • Numerical solver: Random potentials have no closed-form representation or analytic eigenenergies, requiring the network to learn the potential-to-energy mapping from detailed examples.Their generation introduces randomness at multiple stages, and some examples are selected to produce non-trivial wavefunctions.

Deep neural network

The model uses repeated reducing and non-reducing convolutional layers followed by fully connected layers that produce a single predicted quantity. Training uses mean-squared error optimization, with loss no longer decreasing significantly after 1000 epochs.

  • Deep neural network: Reducing convolutional layers use 3 × 3 filters, 64 filters per layer, and stride 2 × 2 to halve image resolution repeatedly.
  • Deep neural network: Non-reducing convolutional layers use unit stride to preserve resolution while adding trainable parameters, and all convolutional layers use ReLU activation.
  • Deep neural network: The architecture combines repeated convolutional units with fully connected layers, ending in a single output used to compute prediction loss.The final convolutional representation feeds a 1024-unit ReLU layer and then a one-output fully connected layer.
  • Deep neural network: AdaDelta with a global learning rate of 0.001 minimizes mean-squared error during training.
  • Deep neural network: After 1000 epochs, the training loss no longer decreases significantly, providing the chosen stopping point for model training.

RESULTS

The convolutional neural network predicts ground-state energies across several potential classes, achieving chemical accuracy on diverse random potentials and extending to excited-state and kinetic-energy targets.

  • Potential classes: The initial models achieved median absolute errors of 1.51 mHa for simple harmonic oscillators, 5.04 mHa for infinite wells, 2.70 mHa for double-well inverted Gaussians, and 2.13 mHa for random potentials.Infinite-well performance was poorer despite simple parameterization, likely because of discontinuities and sparse binary-valued inputs.
  • Random potentials: 1.49 mHa median absolute error places the further-trained random-potential model below the 1.6 mHa chemical-accuracy threshold.The model’s remaining high-energy errors reflect the relative scarcity of high-energy training examples.
  • Transferability: Training on random potentials transferred to double-well inverted Gaussians with a 2.94 mHa error, while an equal mixture of all four classes yielded 5.90 mHa.The mixed-dataset error could be reduced by tuning the architecture for greater dataset variation.
  • Other quantities: The model predicted first excited-state energy with a 10.93 mHa median absolute error, demonstrating applicability beyond ground-state energy.The study trained separate models for ground-state and first excited-state energies.
  • Other quantities: The kinetic-energy model achieved a 2.98 mHa median absolute error despite predicting an expectation value rather than a Hamiltonian eigenvalue.Kinetic energy was evaluated under the numerically computed ground-state wavefunction, and the Hamiltonian and kinetic-energy operators do not commute.
  • Evaluation: Figures 4 and 5 compare true and predicted energies on held-out examples using histograms and error distributions.Figure 4 covers the four potential classes, while Figure 5 covers kinetic and excited-state energy predictions.

CONCLUSIONS

The study presents convolutional deep neural networks as featureless models that learn physical mappings directly from potential data, while highlighting rotational and input-size constraints. It demonstrates mappings from confining potentials to multiple electron observables and argues for broader applicability to spatially encoded problems.

  • CONCLUSIONS: Efficient training and evaluation require uniform input sizes, so future work targets transferability to variable input sizes.This is identified as a notable limitation of the approach.
  • CONCLUSIONS: Augmented training on rotated random potentials gives median absolute errors below 1.6 mHa on original and rotated test sets, while rotated-versus-unaltered predictions differ by 1.7 mHa.The resulting model is not exactly rotationally invariant, but the rotation-related error is reported to be on the same order as the method’s error.
  • CONCLUSIONS: Convolutional neural networks are presented as promising for spatially encoded electronic-structure data, with evaluation lacking the particle-number scaling described for several conventional methods.The authors note that training more complicated systems would be more expensive, but characterize training as a one-time cost.
  • CONCLUSIONS: The model learns mappings from V(r) to ground-state energy, kinetic energy, and first excited-state energy for an electron in a confining 2D well.The authors frame this as learning through examples to rapidly approximate solutions to a set of partial differential equations.

Appendix A: Comparison of machine learning methods

The appendix compares convolutional deep neural networks with kernel ridge regression and random forests on simple harmonic oscillator and random potentials. Traditional methods can excel on simple or small-data cases, whereas the DNN becomes stronger on larger and more varied potentials, with different computational scaling.

  • Appendix A: Comparison of machine learning methods: The comparison evaluates kernel ridge regression and random forests against the convolutional DNN using shared datasets and reported figures for simple harmonic oscillator and random potentials.The kernel ridge regression experiment records training and evaluation walltime together with median absolute error, while the random-forest process is repeated analogously.
  • Appendix A: Comparison of machine learning methods: On simple harmonic oscillator potentials, kernel ridge regression outperforms the DNN with few examples, while both become comparable at larger training sizes and the DNN is slightly better.Kernel ridge regression training scales quadratically, and its fixed-test-size evaluation scales linearly with training-set size; DNN evaluation is unaffected by training-set size.
  • Appendix A: Comparison of machine learning methods: On random potentials, kernel ridge regression performs better with few training examples, but the deep neural network performs much better at larger training-set sizes.Kernel ridge regression barely improves as training size increases while its training walltime increases dramatically.
  • Appendix A: Comparison of machine learning methods: Random forests outperform deep neural networks for every training-set size on the relatively trivial simple harmonic oscillator dataset.Random forests require very long training times, and the plotted training times are scaled by 0.1, making true times ten times larger.
  • Appendix A: Comparison of machine learning methods: On random potentials, random forests perform significantly worse than the deep neural network while also requiring extremely high training time.The comparison is used to support the DNN’s suitability for more varied potentials.

Discussion

The discussion concludes that the DNN is superior except on the simplest potentials, where conventional methods can perform well. Its advantage emerges for more varied inputs, although the timing comparison across algorithms is not quantitatively fair.

  • Discussion: The timing comparison is not quantitatively fair because random forests use one CPU core, kernel ridge regression uses all available cores, and the DNN is highly GPU-parallelized.Error rates are considered quantitatively comparable because KRR and random forests run to convergence, while the DNN is compared at equal walltime.
  • Discussion: For all but the simplest potentials, the deep neural network is vastly superior to kernel ridge regression and random forests.Traditional methods perform well on very simple potentials, where the DNN machinery is unnecessary.
  • Discussion: For more complicated potentials with greater input variation, the deep neural network provides significantly better accuracy in the same amount of time.This contrasts with the simpler-potential regime, where traditional methods can perform well.

Appendix B: Dataset generation

The appendix constructs four classes of two-dimensional confining potentials, including analytically defined wells and highly randomized potentials. Random potentials are designed to be smooth, boundary-constrained, diverse, and energetically suitable for producing varied wavefunctions.

  • Simple Harmonic Oscillator: Simple harmonic oscillator potentials use randomly generated curvatures and centers, with potentials truncated at 20.0 Ha.The potential is defined on a −20 to 20 a.u. grid with 256 × 256 points.
  • Infinite Well: Infinite-well samples are generated by drawing the target energy uniformly from 0 to 0.4 Ha, selecting Lx, and solving for Ly.Invalid combinations are rejected, and Lx and Ly are swapped with 50% probability to avoid systematically elongated wells.
  • Dataset examples: The appendix illustrates the resulting dataset with example random potentials, their ground-state probability densities, and examples from all four potential classes.The selected energy range produces wavefunctions spanning a moderate portion of the spatial domain.
  • Random potentials: The random-potential procedure produces extremely diverse, smooth potentials that approach a maximum of 20.0 at the boundary and span 0–400 mHa.The process uses multiple randomized stages, including binary grids, Gaussian blurring, random masks, exponentiation, and inversion.
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