DeepFRAP: Fast fluorescence recovery after photobleaching data analysis using deep neural networks

Conventional analysis of fluorescence recovery after photobleaching (FRAP) data for diffusion coefficient estimation typically involves fitting an analytical or numerical FRAP model to the recovery curve data using non-linear least squares. Depending on the model this can be time-consuming, especially for batch analysis of large numbers of data sets and if multiple initial guesses for the parameter vector are used to ensure convergence. In this work, we develop a completely new approach, DeepFRAP, utilizing machine learning for parameter estimation in FRAP. From a numerical FRAP model developed in previous work, we generate a very large set of simulated recovery curve data with realistic noise levels. The data is used for training different deep neural network regression models for prediction of several parameters, most importantly the diffusion coefficient. The neural networks are extremely fast and can estimate the parameters orders of magnitude faster than least squares. The performance of the neural network estimation framework is compared to conventional least squares estimation on simulated data, and found to be strikingly similar. Also, a simple experimental validation is performed, demonstrating excellent agreement between the two methods. We make the data and code used publicly available to facilitate further 34development of machine learning-based estimation in FRAP

GMLS-Nets: A framework for learning from unstructured data

Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a non-parametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications

An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions

We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the extreme learning machine (ELM) approach from low to high dimensions. With the first method the unknown solution field in $d$ dimensions is represented by a randomized feed-forward neural network, in which the hidden-layer parameters are randomly assigned and fixed while the output-layer parameters are trained. The PDE and the boundary/initial conditions, as well as the continuity conditions (for the local variant of the method), are enforced on a set of random interior/boundary collocation points. The resultant linear or nonlinear algebraic system, through its least squares solution, provides the trained values for the network parameters. With the second method the high-dimensional PDE problem is reformulated through a constrained expression based on an Approximate variant of the Theory of Functional Connections (A-TFC), which avoids the exponential growth in the number of terms of TFC as the dimension increases. The free field function in the A-TFC constrained expression is represented by a randomized neural network and is trained by a procedure analogous to the first method. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance. These methods can produce accurate solutions to high-dimensional PDEs, in particular with their errors reaching levels not far from the machine accuracy for relatively lower dimensions. Compared with the physics-informed neural network (PINN) method, the current method is both cost-effective and more accurate for high-dimensional PDEs.Comment: 38 pages, 17 tables, 25 figure