Topological Uncertainty for Anomaly Detection in the Neural-network EoS Inference with Neutron Star Data

We study the performance of the Topological Uncertainty (TU) constructed with a trained feedforward neural network (FNN) for Anomaly Detection. Generally, meaningful information can be stored in the hidden layers of the trained FNN, and the TU implementation is one tractable recipe to extract buried information by means of the Topological Data Analysis. We explicate the concept of the TU and the numerical procedures. Then, for a concrete demonstration of the performance test, we employ the Neutron Star data used for inference of the equation of state (EoS). For the training dataset consisting of the input (Neutron Star data) and the output (EoS parameters), we can compare the inferred EoSs and the exact answers to classify the data with the label $k$. The subdataset with $k=0$ leads to the normal inference for which the inferred EoS approximates the answer well, while the subdataset with $k=1$ ends up with the unsuccessful inference. Once the TU is prepared based on the $k$-labled subdatasets, we introduce the cross-TU to quantify the uncertainty of characterizing the $k$-labeled data with the label $j$. The anomaly or unsuccessful inference is correctly detected if the cross-TU for $j=k=1$ is smaller than that for $j=0$ and $k=1$. In our numerical experiment, for various input data, we calculate the cross-TU and estimate the performance of Anomaly Detection. We find that performance depends on FNN hyperparameters, and the success rate of Anomaly Detection exceeds $90\%$ in the best case. We finally discuss further potential of the TU application to retrieve the information hidden in the trained FNN.