Parametrically Managed Activation Function for Fitting a Neural Network Potential with Physical Behavior Enforced by a Low-Dimensional Potential

Abstract

Machine-learned representations of potential energy surfaces generated in the output layer of a feedforward neural network are becoming increasingly popular. One difficulty with neural network output is that it is often unreliable in regions where training data is missing or sparse. Human-designed potentials often build in proper extrapolation behavior by choice of functional form. Because machine learning is very efficient, it is desirable to learn how to add human intelligence to machine-learned potentials in a convenient way. One example is the well-understood feature of interaction potentials that they vanish when subsystems are too far separated to interact. In this article, we present a way to add a new kind of activation function to a neural network to enforce lowdimensional constraints. In particular, the activation function depends parametrically on all of the input variables. We illustrate the use of this step by showing how it can force an interaction potential to go to zero at large subsystem separations without either inputting a specific functional form for the potential or adding data to the training set in the asymptotic region of geometries where the subsystems are separated. In the process of illustrating this, we present an improved set of potential energy surfaces for the 14 lowest 3 A′ states of O 3 . The method is more general than this example, and it may be used to add other low-dimensional knowledge or lower-level knowledge to machine-learned potentials. In addition to the O 3 example, we present a greater-generality method called parametrically managed diabatization by deep neural network (PM-DDNN) that is an improvement on our previously presented permutationally restrained diabatization by deep neural network (PR-DDNN).

Semiclassical Multistate Dynamics for Six Coupled 5A′ States of O + O2

“…In simulations of inelastic collisions in high-enthalpy gases, it has previously been necessary to use qualitative arguments to estimate the probability of electronically excited molecules. − The present results should not be compared directly to experiment since experimental collisions of O 2 ( 3 Δ u ) with O( 3 P) also occur on other symmetry manifolds, and those contributions are not available yet. However, the ability demonstrated here (and in our recent papers , on another symmetry manifold of O + O 2 ) to calculate electronically inelastic cross sections for O + O 2 collisions will allow explicit inclusion of excited electronic states and more realistic simulations in the future.…”

“…In the present study, all biases were set to zero. At the DPEM layer, a parametrically managed activation function is employed, and the 2B potential is added . Therefore, the neural network is effectively fitting the MB potential where F is the parametrically managed activation function, which depends parametrically on the input coordinates where and where f , a , and b are parameters.…”

“…In the present study, all biases were set to zero. At the DPEM layer, a parametrically managed activation function is employed, and the 2B potential is added . Therefore, the neural network is effectively fitting the MB potential bolda j L − 1 = F [ ( w L − 2 ) T a j L − 2 + b L − 2 ] + boldΞ j 2 normalB where F is the parametrically managed activation function, which depends parametrically on the input coordinates F [ z ] = A ( r j max ) B ( r j max ) z where r j max = max ( r j 1 , r j 2 , r j 3 ) and A ( x ) = 0.5 + 0.5 .25em tanh ( f false( x …”

“…We have employed machine learning techniques to circumvent the major difficulties of traditional diabatization methods. − In particular, we developed a series of methods, including diabatization by deep neural network (DDNN), permutationally restrained DDNN (PR-DDNN), and parametrically managed DDNN (PM-DDNN) to perform semiautomatic simultaneous diabatization and fitting. These methods obtain the desired fit to coupled adiabatic potential energy surfaces by generating a DPEM from a set of target adiabatic potential energy surfaces without using any information about orbitals or couplings from the original electronic structure calculations.…”

“…Therefore, the PM-DDNN potential energy surfaces are more accurate in asymptotic regions where subsystems are separated. Recently, we have applied PR-DDNN to 14 coupled electronic states in the 3 A ′ state manifold of the O 2 –O system followed by an improved fit to this problem using PM-DDNN . Here, we apply PM-DDNN to the 5 A ′ state manifold of the O 2 –O system.…”