## 2 Related Work  
The idea of using deep neural networks to emulate physics-based simulations is not completely new. A deep learning framework with graph neural networks is proposed in [7] to simulate complex physical systems involving fluids, rigid solids, and other deformable objects. This idea of building neural network-based emulators for physics-based simulations in different domains--high energy physics, climate science, astrophysics, and seismology has been explored in [8]. The authors propose an algorithm based on neural architecture search to approximate/emulate the simulations using deep neural networks that are accurate and orders of magnitude faster (up to 2 billion times). The main advantage of neural network emulators is that they are orders of magnitude faster and can be useful for extensive parameter exploration and very large scale analysis. Based on our detailed literature survey, we believe this is this is the first work proposing neural network emulators for traffic microscopic simulations to compute measures of effectiveness.  
Machine learning techniques, including deep neural networks have been successfully applied to traffic state data in the past literature. Existing literature on predicting traffic state include either predicting either volumes or performance measures (such as travel times, wait times, queue length, etc.). The prediction horizon could be either at a sub-cycle level or at aggregate intervals (5 min to 1 h). We outline this work below.  
A hybrid method incorporating filtering-based empirical mode decomposition is proposed in [9]. A deep learning-based method with non-parametric regression is proposed in [10], a novel deep neural network architecture with multisegments (recurrent and convolutional layers) is proposed in [11]. A deep generative model based on generative adversarial networks is proposed in [12]. A hybrid method based on linear programming, fuzzy logic, and multi layer perceptron is proposed in [13]. Some of the recent deep learning architectures proposed for spatio-temporal forecasting of traffic state data include temporal graph convolution network [14], spatio temporal residual graph attention network [15], and spatio temporal residual graph attention network [15]. Some relevancy to our work is proposed in [16; 17], which uses real world data to drive the microscopic simulations to compute performance indices such as travel time, emission performance, etc. that are used for the network. The main difference of our work from the existing work is two fold:  
* We propose deep neural networks for estimating the distribution of performance measures instead of doing the microscopic simulations. Our methods are at least four to five orders of magnitude faster;
* Our models can also predict the performance measures for counterfactual signal timing plans, i.e., for a given input traffic and also for different cycle times and green splits (signal timing parameters). This can be useful to study the impact of different signal timing parameters.  
The key observation is that neural network models can be used for computationally efficient parameter exploration. One important application in case of traffic intersections is that this approach can be used to find signal timing parameters for each of the intersection in a city corridor/network that satisfies a particular objective.