Research

In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs  x'(t) = f(t, x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks are universal approximators, we do not need any prior knowledge on the ODE system. Since the model is applied component wise, it can handle systems of any dimension, making it usable for real-world data.

Parameter identification determines the essential system parameters required to build real-world dynamical systems by fusing crucial physical relationships and experimental data. However, the data-driven approach faces main difficulties, such as a lack of observational data, discontinuous or inconsistent time trajectories, and noisy measurements. The ill-posedness of the inverse problem comes from the chaotic divergence of the forward dynamics. Motivated by the challenges, we shift from the Lagrangian particle perspective to the state space flow field's Eulerian description. Instead of using pure time trajectories as the inference data, we treat statistics accumulated from the Direct Numerical Simulation (DNS) as the observable, whose continuous analog is the steady-state probability density function (PDF) of the corresponding Fokker--Planck equation (FPE). We reformulate the original parameter identification problem as a data-fitting, PDE-constrained optimization problem. An upwind scheme based on the finite-volume method that enforces mass conservation and positivity preserving is used to discretize the forward problem. We present theoretical regularity analysis for evaluating gradients of optimal transport costs and introduce three different formulations for efficient gradient calculation. Numerical results using the quadratic Wasserstein metric from optimal transport demonstrate this novel approach's robustness for chaotic dynamical system parameter identification. 

During my time at IPAM as a core participant in the Computational Microscopy Program, I have also started a collaboration with with Jianwei Miao (UCLA), Stanley Osher (UCLA), Minh Pham (UCLA), and Daniel Jacobs (UCLA) on image reconstruction and phase retrieval for biological applications.

In this work we propose to design a Convolutional Neural Network (CNN) to perform in situ coherent diffractive imaging (in situ CDI) for real-time observation of dynamic processes during data acquisition as well as provide a mathematical explanation for the superior results obtained by in situ CDI compared with CDI in the low-dose regime.

Iterative methods for in situ CDI, while powerful, require tuning of several algorithmic parameters and expert strategies. Inspired by the 2022 work of Chang et al. we propose to replace iterative algorithms for in situ CDI with deep learning based methods. In particular we use a CNN trained using only simulated data and exploit the time-invariant static region between the measured diffraction patterns as a powerful real-space constraint; we expect this will result in faster and more robust convergence.

Once fully trained, the CNN can perform real-time phase reconstructions of the structure and dynamics of radiation-sensitive biological materials and would provide a novel, more accurate and fast alternative to iterative algorithms for phase retrieval for applications in medicine, biology and physics.