Fragment Predictions In Proton-Induced Spallation Reaction Based On Bayesian Neural Network

Spallation reaction has attracted considerable interest in various applications,such as aeronautics and astronautics testing,nuclear waste disposal,proton therapy and the experiments based on the third generation of radioactive ion beam facilities.Spallation product cross sections are key infrastructure data for various applications in many aspects.Due to the wide range of incident energy and incident species,furthermore the residue fragments are distributed over a large range of mass and charge numbers,it is a challenge both experimentally and theoretically to obtain accurate and complete energy-dependent spallation cross sections.We apply the Bayesian neural network(BNN)approach to learn measured fragment cross sections of spallation reaction and predict unknowns with uncertainty quantification.The BNN modeling and prediction process is studied.Combining our training data with prior distributions of the model parameters and the vague priors of the hyper-parameters to control the model complexity and avoid overfitting problem.Markov chain Monte Carlo method is applied to optimize the model parameters and obtain the model predictions.We provide two modeling methods,BNN direct fitting method and BNN+SPACS method.BNN direct fitting method is to establish the relationship between the reaction system and the fragment cross section by learning the existing experimental and a few theoretical cross sections.The network structure of 7-35-1 in this work is adopted,that is 7 inputs x_i={A_p,Z_p,E,A,Z,N-Z,B_p},single hidden layer with 35 hidden neurons and one output t_i=lg?(?is the fragment cross section).Comparing with the experimental data,it was demonstrated that the BNN direct fitting method provides a good prediction of the isotopic cross-sections in proton induced spallation reactions,in particular for fragments with small charge number.BNN+SPACS method is to make up for the defect of theoretical model by learning the residuals between experimental and theoretical data.The network structure of 5-23-1 is adopted,that is 5 inputs xi={Ap,Zp,E,A,Z},single hidden layer with 23 hidden neurons and one output.The predictions of BNN+SPACS method is verified to well reproduced the experimental data and has a good ability of generalization.It is demonstrated that the BNN method is particularly useful for evaluations of spallation cross sections when incomplete experimental data are available.The BNN evaluation results are quite satisfactory on distribution positions and energy dependencies of spallation cross sections.These two methods will have practical applications in nuclear astrophysics,ADS and proton therapy,etc.