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Bayesian approach to model-based extrapolation of nuclear observables

Léo Neufcourt, Yuchen Cao, Witold Nazarewicz, Frederi Viens

arXiv:1806.00552v3nucl-thstat.ML

TL;DR

Extrapolating nuclear observables is necessary where measurements are unavailable, including in astrophysical regions far from stability. The paper applies Bayesian Gaussian processes and neural networks to model residuals and quantify uncertainty. Both improve predictive accuracy, with Gaussian processes performing more stably and the uncertainty estimates tracking their nominal coverage levels.

  • Problem

    Many astrophysical nuclei lie beyond experimental reach, requiring theoretical predictions in regions where masses have not been measured.

    Source: §I · pp. 1–2

  • Method

    The study applies fully Bayesian Gaussian processes and neural networks to even-even nuclei’s two-neutron separation-energy residuals and predictions.

    Source: §I · p. 2, §VI · p. 16

  • Results

    Both approaches significantly reduce rms deviations, but Gaussian processes perform better and more stably, while empirical coverage probabilities match reference values well.

    Source: §VI · p. 16

  • Takeaways & Limitations

    Gaussian-process-assisted extrapolations for four global models give internally consistent tin-isotope predictions within their estimated credibility intervals.

    Source: §VI · p. 16

    Without experimental data testing Bayesian neural networks far from stability, the value of their corrections there remains unknown.

    Source: §VI · p. 16

Abstract

from arXiv · show

The mass, or binding energy, is the basis property of the atomic nucleus. It determines its stability, and reaction and decay rates. Quantifying the nuclear binding is important for understanding the origin of elements in the universe. The astrophysical processes responsible for the nucleosynthesis in stars often take place far from the valley of stability, where experimental masses are not known. In such cases, missing nuclear information must be provided by theoretical predictions using extreme extrapolations. Bayesian machine learning techniques can be applied to improve predictions by taking full advantage of the information contained in the deviations between experimental and calculated masses. We consider 10 global models based on nuclear Density Functional Theory as well as two more phenomenological mass models. The emulators of S2n residuals and credibility intervals defining theoretical error bars are constructed using Bayesian Gaussian processes and Bayesian neural networks. We consider a large training dataset pertaining to nuclei whose masses were measured before 2003. For the testing datasets, we considered those exotic nuclei whose masses have been determined after 2003. We then carried out extrapolations towards the 2n dripline. While both Gaussian processes and Bayesian neural networks reduce the rms deviation from experiment significantly, GP offers a better and much more stable performance. The increase in the predictive power is quite astonishing: the resulting rms deviations from experiment on the testing dataset are similar to those of more phenomenological models. The empirical coverage probability curves we obtain match very well the reference values which is highly desirable to ensure honesty of uncertainty quantification, and the estimated credibility intervals on predictions make it possible to evaluate predictive power of individual models.

I. INTRODUCTION

Reliable nuclear-mass predictions matter because many astrophysical nuclei lie beyond experimental reach, where models must extrapolate. This paper investigates whether Bayesian emulation of model residuals can improve predictions and quantify their uncertainty.

I. INTRODUCTION

  • Nuclear masses shape stability, decay channels, and reaction rates, yet astrophysical r-process nuclei can be too short-lived for foreseeable measurement.Source: §I · p. 1
  • Reliable extrapolations are challenging because a model’s calibration error does not necessarily predict its ability to describe new masses.Source: §I · p. 2
  • The study tests global models on even-even nuclei using two-neutron separation energies, a lower-residual observable that helps locate the neutron dripline.Source: §I · p. 2
  • Bayesian emulators estimate model residuals from known masses, then combine those corrections with theoretical predictions to estimate unknown separation energies and their uncertainty.Source: §I · p. 2

II. NUCLEAR MODELS

The study compares global mass models and tests residual-based predictions using measured separation energies. Its datasets support tests on later measurements, while DFT predictions stop at the formal two-neutron dripline.

II. NUCLEAR MODELS

  • The evaluated models comprise FRDM-2012 and HFB-24, six Skyrme-DFT models, and four covariant-DFT models.Source: §II · p. 3
  • DFT predictions end when the neutron chemical potential becomes positive, the condition under which the standard localized-density solution is not meaningful.Source: §II · p. 3
  • Training uses 537 AME2003 separation energies; 55 later AME2016 measurements test predictions, with four JYFLTRAP measurements used for a further test.Source: §II · p. 3
  • Residuals show long-range patterns and substantial deviations around neutron magic numbers, with the magnitude and sign of trends depending on the model.Source: §II · p. 3

III. STATISTICAL METHODOLOGY

The statistical methodology uses fully Bayesian Gaussian-process and neural-network models to infer residuals and make predictions with uncertainty. Posterior distributions incorporate parameter uncertainty, while the chosen prior framework is limited by the absence of external data.

III. STATISTICAL METHODOLOGY

  • Gaussian processes can exploit short-range correlations, while Bayesian neural networks are expected to capture long-range trends in residuals.Source: A. Bayesian approach · p. 4
  • The Bayesian model relates proton and neutron numbers to observed residuals through a function with unknown parameters and an additive noise term.Source: A. Bayesian approach · p. 4
  • Bayes’ theorem combines the likelihood and prior to form parameter posteriors, which are integrated to predict residuals where measurements are unavailable.Source: A. Bayesian approach · p. 4
  • A credibility interval is an interval containing a specified posterior probability for a parameter or predicted value.Source: A. Bayesian approach · p. 5
  • Without external data to inform priors, the analysis uses a single-level framework; posterior samples are generated with 100,000 Metropolis-Hastings iterations.Source: A. Bayesian approach · p. 6

B. Predictions and uncertainties

Posterior samples provide residual corrections and uncertainty estimates for predicted separation energies. Credibility intervals can be constructed from these samples, with normality-based shortcuts used after the authors checked approximate normality.

B. Predictions and uncertainties

  • The posterior mean of predicted residual samples supplies the correction added to the theoretical separation energy, and their sample standard deviation gives the one-sigma error bar.Source: B. Predictions and uncertainties · p. 6
  • The emulator’s uncertainty output supports measures beyond predictive accuracy, including credibility intervals built from posterior samples.Source: B. Predictions and uncertainties · p. 6
  • For approximately normal predicted residuals, the authors use standard-deviation shortcuts for two-sided 95% and 68% credibility intervals.Source: B. Predictions and uncertainties · p. 6
  • The authors checked approximate normality before using the shortcut; more generally, intervals can instead contain a specified fraction of consecutive ordered posterior samples.Source: B. Predictions and uncertainties · p. 6

C. Objective and evaluation

Predictive performance is assessed on nuclei outside the training domain, reflecting the goal of extrapolating toward more unstable nuclei. The evaluation also checks whether uncertainty intervals achieve their stated coverage on test data.

C. Objective and evaluation

  • Model performance is judged by rms-error improvement on testing nuclei outside the training domain, rather than by randomly held-out interior points.The criterion targets predictive power toward more unstable nuclei.Source: C. Objective and evaluation · p. 7
  • The evaluation considers both predictive accuracy and whether each model accurately estimates its own uncertainty.Rms-error improvement alone does not assess uncertainty quantification.Source: C. Objective and evaluation · p. 7
  • Empirical coverage probability is the share of testing observations inside credibility intervals at a specified nominal level.Honest uncertainty quantification should make empirical coverage close to the nominal level; substantially lower or higher coverage indicates intervals that are too narrow or too wide, respectively.Source: C. Objective and evaluation · p. 7

A. Gaussian Process model

The GP models residuals as a spatially correlated Gaussian field over the nuclear chart, while the BNN uses a one-hidden-layer neural network with Bayesian parameter inference. Their specifications differ in how they represent dependence and uncertainty.

A. Gaussian Process model

  • The GP treats residuals over the discrete (Z, N) nuclear domain as a zero-mean Gaussian field with an exponential-quadratic covariance kernel.The kernel represents spatial dependence between neighboring nuclei.Source: A. Gaussian Process model · p. 7
  • The GP kernel parameters encode dependence strength and characteristic correlation ranges in the proton and neutron directions.The kernel is calibrated on observed residuals and used for conditional predictions; its noise scale is fixed at 0.0235 MeV to represent experimental uncertainty.Source: A. Gaussian Process model · p. 7

B. Bayesian Neural Network

  • The BNN uses a single hidden layer with 30 neurons, chosen with statistical parsimony in view of the limited data.The paper defines it as a Bayesian analysis of an artificial neural network with additive noise, estimating posterior distributions for model parameters and predictions.Source: B. Bayesian Neural Network · p. 8
  • In the BNN, the function is deterministic and independent, identically distributed Gaussian noise is assumed for observations.The resulting predictions are conditionally independent of training data given the model parameters and noise scale.Source: B. Bayesian Neural Network · p. 8
  • The BNN uses hierarchical weight priors, while its activation choice can matter for extrapolation on the bounded nuclear domain.The paper notes that tanh has linear tails and suggests a more local activation, such as a Gaussian kernel, may be more appropriate for globally modeled residuals.Source: B. Bayesian Neural Network · p. 8
  • With about 500 data points, the study limits the BNN to one hidden layer; adding another layer decreases performance and adds 120 parameters.The authors attribute the constraint to the small dataset and note that the GP has far fewer parameters, giving it an informal UQ advantage in this study.Source: B. Bayesian Neural Network · p. 8

C. Refinements

For extrapolation, the basic GP improves predictive power while the basic BNN performs poorly at noise reduction; the reported refinements are especially consequential for the BNN. The GP parameter posteriors are described as well determined, with correlation effects concentrated over a specific neighborhood.

C. Refinements

  • The basic GP significantly improves predictive power, whereas the basic BNN performs poorly at noise reduction in extrapolation.The paper distinguishes these results from prior BNN successes assessed with traditional cross-validation rather than extrapolation error.Source: C. Refinements · p. 9
  • Prior studies’ reduced-domain comparisons excluded light nuclei, while this work also implements that restriction as GP(H) and BNN(H).Those studies used different domain restrictions, including isotopic chains above 40Ca or nuclei with Z and N above 8 and experimental errors below 100 keV.Source: C. Refinements · p. 9
  • Adding magic-gap proximity features is determining for BNN performance and brings only a minor improvement to GP.The four-dimensional input uses distances to nearby magic proton and neutron numbers and a promiscuity factor.Source: C. Refinements · p. 9
  • All three DD-PC1 GP parameters have relatively narrow posterior densities; about 90% of correlation effects lie within Z ± 4 and N ± 2.The parameter η is 0.87 MeV; Fig. 4 displays posterior distributions with posterior means and standard deviations.Source: C. Refinements · p. 9, Figure 4 · p. 9

V. RESULTS

On the AME2016–AME2003 testing set, both GP and BNN corrections reduce S2n rms residuals, with GP performing significantly better. Corrected predictions improve residual patterns, and empirical coverage curves closely track their reference values for most models.

A. Training set: AME2003, testing set: AME2016

B. Training set AME2016-H, testing set JYFLTRAP data

On the four JYFLTRAP measurements, GP reduces rms residuals significantly, while BNN is ineffective for several models. Predictions trained on AME2003-H and AME2016-H differ insignificantly.

B. Training set AME2016-H, testing set JYFLTRAP data

  • GP significantly reduces rms residuals on JYFLTRAP data, while BNN worsens results for SLy4, SkP, SV-min, UNEDF0, and DD-MEδ.The underlying model rms residuals are 0.11–0.16 MeV for several recently developed EDFs and 0.13 MeV for FRDM-2012 and HFB-24.Source: B. Training set AME2016-H, testing set… · p. 12, Table II · p. 12
  • Predictions from AME2003-H and AME2016-H training datasets differ insignificantly on the JYFLTRAP data.The authors report that recent measurements do not constrain predictions enough to modify them in a smooth mass-surface region.Source: B. Training set AME2016-H, testing set… · p. 12

C. Extrapolations

The authors use validated Bayesian credibility intervals to examine extrapolations, while noting that their testing points were relatively close to the training domain. For Sn, GP and BNN give different dripline estimates, and the authors recommend a one-sided 1.65-sigma interval as a lower limit.

C. Extrapolations

  • The extrapolation validation is bounded because the testing points were relatively close to the training domain.The authors frame the validated credibility intervals as a basis for extrapolation trust, subject to this boundary.Source: C. Extrapolations · pp. 12–13
  • For Sn, GP and BNN extrapolations have overlapping credibility intervals in most cases, while their interval sizes increase farther from known data.The empirical 136,138Sn values from Ref. are well reproduced by both approaches.Source: C. Extrapolations · p. 13
  • HFB-24’s close fit to the training data makes its emulator blind to underlying uncertainty outside that domain, producing underestimated uncertainty and small extrapolation intervals.The authors suggest incorporating an additional error term for predictions outside the training domain.Source: C. Extrapolations · p. 13
  • For DD-PC1, the posterior-mean dripline estimate is N*=126 with GP(T+H) and N*=118 with BNN(T+H); the recommended 1.65-sigma lower limits are N*=118 and N*=104, respectively.The authors recommend the neutron number where the 1.65-sigma interval crosses S2n = 0 as a lower limit, corresponding to at least 95% probability that the dripline is there or below.Source: C. Extrapolations · p. 13

D. Numerical considerations

The GP emulator reaches a stationary regime with fewer tuning samples than the BNN, whose extrapolative predictions can vary across runs. Improving BNN stability would require impractical computational effort, while GP numerical issues are largely absent despite higher evaluation time.

D. Numerical considerations

  • Even at 1,000,000 MCMC samples, DD-PC1+BNN(T+H) runs can differ significantly in Sn predictions and confidence intervals.The comparison uses two runs with 100,000 samples after 10,000-sample tuning and two with 1,000,000 after 100,000-sample tuning.Source: D. Numerical considerations · p. 13, Figure 10 · p. 15
  • Improving BNN convergence would require two orders of magnitude more simulations, which the authors judge computationally impractical.NUTS and Metropolis achieved similar convergence for comparable computation time in these simulations.Source: D. Numerical considerations · p. 15
  • GP numerical issues are essentially absent with three parameters versus over one hundred for BNN, though GP prediction sampling takes longer than BNN evaluation.The GP sampling cost is still far below the factor of 100 that would be needed to improve BNN stability.Source: D. Numerical considerations · p. 15
  • 10,000 tuning samples put GP predictions and uncertainties in a stationary regime, versus about 100,000 for BNN.The figure tracks posterior means and standard deviations for several Sn isotopes as a function of MCMC sample count.Source: D. Numerical considerations · p. 14, Figure 11 · p. 15

VI. CONCLUSIONS

Bayesian GP and BNN corrections use residual patterns to improve extrapolations of nuclear S2n, with GP producing better, more stable test performance. The study evaluates uncertainty coverage and discusses applications and boundaries of distant predictions.

VI. CONCLUSIONS

  • For AME2016-2003, GP T+H yields 400-500 keV rms deviations in S2n; both GP and BNN reduce errors, but GP performs better and more stably.For the testing dataset, corrected UNEDF0, UNEDF1, and SV-min predictions reach quality similar to more phenomenological models.Source: §VI · p. 16
  • GP builds local corrections suited to residual trends around magic gaps, while BNN's sigmoid activation has linear tails that are poorly suited to bounded extrapolation.The residuals show appreciable local structure around magic gaps but no global structure; the authors say GP corrections far beyond training are negligible by design.Source: §VI · p. 16
  • BNN performance on long-range extrapolations cannot be assessed without supporting experimental data far from the stability range.The authors identify uncertainty quantification from credibility intervals as the main interest of Bayesian methods for distant extrapolations.Source: §VI · p. 16
  • Empirical coverage probability curves match reference values very well and are slightly conservative in most cases, supporting honest uncertainty quantification without wasteful intervals.Source: §VI · p. 16
  • For the JYFLTRAP region, new unstable-nucleus data had minor impact on predictive power, a result the authors attribute to a fairly smooth mass surface.They caution against generalizing this outcome and propose testing boundary-region experiments with synthetic data and quantified uncertainty to assess cost versus risk.Source: §VI · p. 16
  • Although HFB-24 and FRDM-2012 reproduce measured masses better, their extrapolations into the neutron-rich r-process region are similar to those of well-optimized DFT models.The authors caution that mean values alone tell only part of the story and discuss combining credibility intervals to reduce uncertainty regions.Source: §VI · p. 16
  • Figure 12 shows four GP(T+H)-corrected Sn-chain predictions that are consistent overall when statistical corrections and uncertainty are included.Source: §VI · p. 16, Figure 12 · p. 17
  • The Bayesian strategy can be applied to other observables, including one-neutron separation energies and beta-, alpha-, and fission-decay Q-values, to quantify uncertainties in mass differences.Source: §VI · p. 17
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