Research
Reliability of surrogate models based on operator learning of parametric partial differential equations
In this work, we aim to understand and improve the reliability of neural operators as surrogates of parametric nonlinear PDEs in infinite-dimensional Bayesian inverse problems (BIPs). We first derive an a-priori bound that allows us to understand how the error in operator learning controls the error in the posterior distribution of BIPs. We then proposed a post-training error correction strategy. This strategy improves the accuracy of a trained neural operator by solving a linear variational problem based on the predictions of the neural operator. We show that such a correction step leads to a quadratic reduction of the approximation error of well-trained neural operators.
Predictions before collecting data
Collected microscopy data
Predictions after collecting data
Bayesian predictive modeling of diblock copolymer thin film self-assembly
In a series of works, we formulated and proposed suitable methodologies for the Bayesian calibration of models for diblock copolymer self-assembly. In particular, the model calibration procedure accounts for the aleatoric randomness represented by the metastability of the self-assembly. We advocate using likelihood-free inference methodologies in adjunct to constructing summary statistics for effective parameter inference. We explored inference methodologies such as pseudo-marginal methods and triangular transport maps. We designed summary statistics based on power spectrum and energy functionals from microscopy data. The utilities of these summary statistics are quantified by estimating expected information gains.
Measure transport: [preprint]
PDE-constrained optimization algorithms for polymer self-consistent field calculations
In this work, we formulate and analyze self-consistent field (SCF) calculations of diblock copolymers in the framework of PDE-constrained optimization. We derive the Hessian action of the SCF optimization and show that the semi-implicit Seidel (SIS) scheme proposed by Ceniceros and Fredrickson [link] assumes a particular block diagonal Hessian approximation. We extend the SIS scheme from a Fourier-based scheme to a real space–based one utilizing Laplacian operators. This extension allows us to accelerate SCF optimization on domains with complex geometries.
[thesis, chap. 3]
Figure: Superior convergence speed in SCF calculations for diblock copolymer thin films with strong immiscibility.
Figure: 3D simulation of diblock copolymer thin films with quadratic reduction of the residual norm value. (right bottom).
Energy-stable mass-conservative Newton iterations for polymer density functional theory calculations
In this work, we analyzed and proposed a fast and robust algorithm for directly minimizing the Ohta–Kawasaki free energy. A mass-conservative Newton iteration for energy minimization is formulated. We utilize an adaptive Gauss–Newton convexification of the Hessian operator and inexact line search to ensure that the iteration is guaranteed to be monotonically energy descending. The algorithm is used to study the effects of polymer–substrate interactions in chemoepitaxy-directed self-assembly of diblock copolymers (see figures).