Peter Y. Lu

Selected Publications

1. Extracting Interpretable Physical Parameters from Spatiotemporal Systems Using Unsupervised Learning

   Lu, Peter Y., Kim, Samuel, and Soljačić, Marin

   Phys. Rev. X 2020

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   Experimental data are often affected by uncontrolled variables that make analysis and interpretation difficult. For spatiotemporal systems, this problem is further exacerbated by their intricate dynamics. Modern machine learning methods are particularly well suited for analyzing and modeling complex datasets, but to be effective in science, the result needs to be interpretable. We demonstrate an unsupervised learning technique for extracting interpretable physical parameters from noisy spatiotemporal data and for building a transferable model of the system. In particular, we implement a physics-informed architecture based on variational autoencoders that is designed for analyzing systems governed by partial differential equations. The architecture is trained end to end and extracts latent parameters that parametrize the dynamics of a learned predictive model for the system. To test our method, we train our model on simulated data from a variety of partial differential equations with varying dynamical parameters that act as uncontrolled variables. Numerical experiments show that our method can accurately identify relevant parameters and extract them from raw and even noisy spatiotemporal data (tested with roughly 10% added noise). These extracted parameters correlate well (linearly with $R^2>0.95$) with the ground truth physical parameters used to generate the datasets. We then apply this method to nonlinear fiber propagation data, generated by an ab initio simulation, to demonstrate its capabilities on a more realistic dataset. Our method for discovering interpretable latent parameters in spatiotemporal systems will allow us to better analyze and understand real-world phenomena and datasets, which often have unknown and uncontrolled variables that alter the system dynamics and cause varying behaviors that are difficult to disentangle.

2. Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery

   Kim, Samuel, Lu, Peter Y., Mukherjee, Srijon, Gilbert, Michael, Jing, Li, Čeperić, Vladimir, and Soljačić, Marin

   IEEE Transactions on Neural Networks and Learning Systems 2020

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   Symbolic regression is a powerful technique to discover analytic equations that describe data, which can lead to explainable models and the ability to predict unseen data. In contrast, neural networks have achieved amazing levels of accuracy on image recognition and natural language processing tasks, but they are often seen as black-box models that are difficult to interpret and typically extrapolate poorly. In this article, we use a neural network-based architecture for symbolic regression called the equation learner (EQL) network and integrate it with other deep learning architectures such that the whole system can be trained end-to-end through backpropagation. To demonstrate the power of such systems, we study their performance on several substantially different tasks. First, we show that the neural network can perform symbolic regression and learn the form of several functions. Next, we present an MNIST arithmetic task where a convolutional network extracts the digits. Finally, we demonstrate the prediction of dynamical systems where an unknown parameter is extracted through an encoder. We find that the EQL-based architecture can extrapolate quite well outside of the training data set compared with a standard neural network-based architecture, paving the way for deep learning to be applied in scientific exploration and discovery.