## Prediction of nuclear masses with an artificial neural network

LA-UR-22-24945

### Matthew Mumpower

Nuclear Structure

Friday June 17$^{th}$ 2022

Theoretical Division

## ML/AI: Text → image

A small cactus wearing a straw hat and neon sunglasses in the Sahara desert.

## ML/AI: Text → bling

A bucket bag made of blue suede. The bag is decorated with intricate golden paisley patterns. The handle of the bag is made of rubies and pearls.

## ML/AI: person generation

Every refresh of this page generates a new person (who does not exist)

Generative Adversarial Network (GAN) • http://thispersondoesnotexist.com

## Nuclear physics from a forward problem perspective

When we model nuclear properties we think of it as a 'forward' problem

We start with our model and parameters and try to match measurements / observations

This is an extremely successful approach and is generally how we use theory models

Mathematically: $f(\vec{\boldsymbol{x}}) = \vec{\boldsymbol{y}}$

Where $f$ is our model, $\vec{\boldsymbol{x}}$ are the parameters for the model and $\vec{\boldsymbol{y}}$ are the predictions

Figure by Mumpower

## Forward problem... problems...

There can be many challenges with this approach

For instance: model computationally expensive or how do we update our model if we don't match data?

Or perhaps...

We don't always know what modifications are required or the physics could be very hard if not impossible to model given current computational limitations (e.g. many-body problem)

Treating this as an 'inverse problem' provides an alternative approach...

Figure by Mumpower

## Nuclear physics from an inverse problem perspective

We can then try to figure out what models may match relevant observations

This approach allows us to vary and optimize the model!

Mathematically, still: $f(\vec{\boldsymbol{x}}) = \vec{\boldsymbol{y}}$; but we want to now find $f$; fixing $\vec{\boldsymbol{x}}$

Machine learning and artificial intelligence algorithms are ideal for this class of problems...

Figure by Mumpower

## How do we find $f$? An example...

My son Zachary has been very keen on learning his letters from an early age

Letters are well known (this represents the data we want to match to)

But the model (being able to draw - eventually letters) must be developed, practiced and finally optimized

This is a complex, iterative process - it takes times to find '$f$'!

## Finding $f$... a first attempt

Example from 1 years old; random drawing episode; time taken: 10 minutes

Drawing by Z. R. Mumpower

## Finding $f$... okay for real this time

Example from 2.5 years old; first ever attempt to draw letters / alphabet; time taken: 2 hours!

Drawing by Z. R. Mumpower

## Finding $f$... now we're getting somewhere

Example from 3.5 years old; drawing scenery; time taken: 10 minutes

Drawing by Z. R. Mumpower

## Finding $f$... something legible!

Example from 4 years old; tracing the days of the week; time taken: 30 minutes

Drawing by Z. R. Mumpower

## Understanding of nuclear astrophysics requires many nuclear properties

Nucleosynthesis masses, $\beta$-decay rates, reaction rates & branching ratios

Mumpower et al. PPNP 86 (2016) • Horowitz et al. J Phys G 083001 (2019) • Kajino et al. PPNP 107 (2019) • Cowan et al. RMP 93 (2021)

## Let's apply this idea to a basic property...

A relatively easy choice is a scalar property; how about atomic binding energies (masses)?

But... writing down the many-body quantum mechanical Hamlitonian is hard; want to find ground state

Recall: $f(\vec{\boldsymbol{x}}) = \vec{\boldsymbol{y}}$

$f$ will be a neural network (a model that can change)

$\vec{\boldsymbol{y}_{obs}}$ are the observations (e.g. Atomic Mass Evaluation) we want to match $\vec{\boldsymbol{y}}$ with

But what about $\vec{\boldsymbol{x}}$?

Why not a physically motivated feature space!?

proton number ($Z$), neutron number, ($N$), nucleon number ($A$), etc.

Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Network feature spaces ($\vec{\boldsymbol{x}}$)

We use a variety of possible physically motivated features as inputs into the model

Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Mixture Density Network

We take a Bayesian approach: our network inputs are sampled based on nuclear data uncertainties

Our network outputs are therefore statistical

We can represent outputs by a collection of Gaussians (for masses we only need 1)

Our network also provides a well-quanitfied estimate of uncertainty (fully propagated through the model)

Network topology: fully connected $\sim$4-8 layers; 8 nodes; regularization via weight-decay

Last layer converts output to Gaussian ad-mixture

Bishop (1994 & 2006) • Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Results: modeling atomic masses

Here we show results along the lead (Pb) [$Z=82$] isotopic chain

Data from the Atomic Mass Evaluation (AME2016) in black stars

Increasing sophistication of the model feature space indicated by M##

Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Results: modeling atomic masses

Results across the chart of nuclides with increasing feature space complexity

Note the decrease in the maximum value of the scale

Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Results: modeling atomic masses

A summary of results

Some lessons learned:

More physics added to the feature space → the better our description of atomic masses

Attempting to fit and predict other mass related quantities (e.g. separation energies) does impact our results

Although marginal compared to choice of feature space

Lovell (FRIB-TA early career award winner!), Mohan, Sprouse & Mumpower arXiv:2201.00676 (2022)

## Training on mass data

We train on mass data using a log-loss, $\mathcal{L}_1$

Training takes a considerably long time

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Introduce physical constraint via loss function

Now we optimize data, $\mathcal{L}_1$, and introduce a second (physical) constraint, $\mathcal{L}_2$

We have tried many physics-based losses, this particular one (Garvey-Kelson relations) performs well

Garvey & Kelson PRL 16 197 (1966) • Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Total loss: sum the two losses

$\mathcal{L}_\textrm{total} = \mathcal{L}_1 + \lambda_\textrm{phys} \mathcal{L}_2$ ; $\lambda_\textrm{phys}=1$ in this case

Good match to data does not imply good physics (although by happenstance it is in the end in this case)!

Garvey & Kelson PRL 16 197 (1966) • Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Mass trends along Nd isotopic chain, $Z=60$

Results plotted relative to the Los Alamos FRDM model

Modern mass models draw a 'straight line' through data

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Results of new model Nd, $Z=60$

Probabilistic network → well-defined uncertainties predicted for every nucleus ( 1$\sigma$, 2$\sigma$ 3$\sigma$)

Trends in data are captured far better than any mass model

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Results of new model Nd, $Z=60$

New data (unknown to the model) are in agreement with the MDN model predictions

Model is now trained on only 20% of the AME and predicting 80%!

The training set is random; different datasets perform differently

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## Results of new model In, $Z=49$

Our results are not limited to any mass region - all of them are described equally well ($Z \geq 20$)

Results are better than any existing mass model; uncertainties inherently increase with extrapolation

We can predict new measurements and capture trends (important for astrophysics)

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## How can we interpret these results?

One method is to calculate Shapley values (Shapely 1953 • Lundberg & Lee 2017)

Idea arises in Game Theory when a coalition of players (ML - features) optimize gains relative to costs

The players may contribute unequally, but the payoff is mutually beneficial

Applicable to many applications, ML in particular to understand feature importance

Lipovetsky, Stan, and Michael Conklin, Applied Stochastic Models in Business and Industry (2001) • Strumbelj, Erik, and Kononenko, Knowledge and information systems (2014) • Figure from SHAP Python package (https://shap.readthedocs.io/)

## Ranking feature importance

SHAP calculated for all of the AME (2020) [predictions!]

The model behaves like a theoretical mass model

Macroscopic variables most important followed by quantum corrections

Mumpower, Sprouse, Lovell, & Mohan arXiv:2203.10594 (2022)

## What happens when there's new data? (generalization)

The network is able to incorporate this knowledge and updated predictions via Bayes' theorem

New information provided at closed shell $N=126$

Mumpower et al. in preparation (2022)

## Special thanks to

My collaborators

A. Lovell, A. Mohan, & T. Sprouse

Postdoc

And to my son Zachary for the use of his drawings

Los Alamos National Laboratory Caveat

The submitted materials have been authored by an employee or employees of Triad National Security, LLC (Triad) under contract with the U.S. Department of Energy/National Nuclear Security Administration (DOE/NNSA).

Accordingly, the U.S. Government retains an irrevocable, nonexclusive, royaltyfree license to publish, translate, reproduce, use, or dispose of the published form of the work and to authorize others to do the same for U.S. Government purposes.

## Summary

We have developed an interpretable neural network capable of predicting masses

We can predict masses accurately and suss out trends in data

Our network is probabilistic, each prediction has an associated uncertainty

Our model can extrapolate and retains the physics

The quality of the training data is important (not shown)

We can classify the information content of existing and future measurements with this type of modeling

We are now ready to study the impact of this modeling in astrophysics!

Results / Data / Papers @ MatthewMumpower.com