Peter Y. Lu

Selected Publications

1. Discovering Dynamical Parameters by Interpreting Echo State Networks

   Oreoluwa Alao*, Peter Y. Lu*, and Marin Soljačić

   NeurIPS 2021 AI for Science Workshop (2021) — Best Paper Award

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   Reservoir computing architectures known as echo state networks (ESNs) have been shown to have exceptional predictive capabilities when trained on chaotic systems. However, ESN models are often seen as black-box predictors that lack interpretability. We show that the parameters governing the dynamics of a complex nonlinear system can be encoded in the learned readout layer of an ESN. We can extract these dynamical parameters by examining the geometry of the readout layer weights through principal component analysis. We demonstrate this approach by extracting the values of three dynamical parameters (σ, ρ, β) from a dataset of Lorenz systems where all three parameters are varying among different trajectories. Our proposed method not only demonstrates the interpretability of the ESN readout layer but also provides a computationally inexpensive, unsupervised data-driven approach for identifying uncontrolled variables affecting real-world data from nonlinear dynamical systems.

2. Discovering Sparse Interpretable Dynamics from Partial Observations

   Peter Y. Lu, Joan Ariño, and Marin Soljačić

   (2021)

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   Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. We propose a machine learning framework for discovering these governing equations using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model. Our tests show that this method can successfully reconstruct the full system state and identify the underlying dynamics for a variety of ODE and PDE systems.

3. Scalable and Flexible Deep Bayesian Optimization with Auxiliary Information for Scientific Problems

   Samuel Kim, Peter Y. Lu, Charlotte Loh, Jamie Smith, Jasper Snoek, and Marin Soljačić

   (2021)

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   Bayesian optimization (BO) is a popular paradigm for global optimization of expensive black-box functions, but there are many domains where the function is not completely black-box. The data may have some known structure, e.g. symmetries, and the data generation process can yield useful intermediate or auxiliary information in addition to the value of the optimization objective. However, surrogate models traditionally employed in BO, such as Gaussian Processes (GPs), scale poorly with dataset size and struggle to incorporate known structure or auxiliary information. Instead, we propose performing BO on complex, structured problems by using Bayesian Neural Networks (BNNs), a class of scalable surrogate models that have the representation power and flexibility to handle structured data and exploit auxiliary information. We demonstrate BO on a number of realistic problems in physics and chemistry, including topology optimization of photonic crystal materials using convolutional neural networks, and chemical property optimization of molecules using graph neural networks. On these complex tasks, we show that BNNs often outperform GPs as surrogate models for BO in terms of both sampling efficiency and computational cost.

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

   Peter Y. Lu, Samuel Kim, and Marin Soljačić

   Physical Review 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.

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

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

   IEEE Transactions on Neural Networks and Learning Systems (2021)

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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.