Dongwei Ye

Dongwei Ye, Valeria Krzhizhanovskaya, Alfons G. Hoekstra

Parametric reduced-order modelling often serves as a surrogate method for hemodynamics simulations to improve the computational efficiency in many-query scenarios or to perform real-time simulations. However, the snapshots of the method require to be collected from the same discretisation, which is a straightforward process for physical parameters, but becomes challenging for geometrical problems, especially for those domains featuring unparameterised and unique shapes, e.g. patient-specific geometries. In this work, a data-driven surrogate model is proposed for the efficient prediction of blood flow simulations on similar but distinct domains. The proposed surrogate model leverages group surface registration to parameterise those shapes and formulates corresponding hemodynamics information into geometry-informed snapshots by the diffeomorphisms constructed between a reference domain and original domains. A non-intrusive reduced-order model for geometrical parameters is subsequently constructed using proper orthogonal decomposition, and a radial basis function interpolator is trained for predicting the reduced coefficients of the reduced-order model based on compressed geometrical parameters of the shape. Two examples of blood flowing through a stenosis and a bifurcation are presented and analysed. The proposed surrogate model demonstrates its accuracy and efficiency in hemodynamics prediction and shows its potential application toward real-time simulation or uncertainty quantification for complex patient-specific scenarios.

Konstantinos Kevopoulos, Dongwei Ye

Surrogate modelling is widely applied in computational science and engineering to mitigate computational efficiency issues for the real-time simulations of complex and large-scale computational models or for many-query scenarios, such as uncertainty quantification and design optimisation. In this work, we propose a parametric framework for kernel-based dynamic mode decomposition method based on the linear and nonlinear disambiguation optimization (LANDO) algorithm. The proposed parametric framework consists of two stages, offline and online. The offline stage prepares the essential component for prediction, namely a series of LANDO models that emulate the dynamics of the system with particular parameters from a training dataset. The online stage leverages those LANDO models to generate new data at a desired time instant, and approximate the mapping between parameters and the state with the data using deep learning techniques. Moreover, dimensionality reduction technique is applied to high-dimensional dynamical systems to reduce the computational cost of training. Three numerical examples including Lotka-Volterra model, heat equation and reaction-diffusion equation are presented to demonstrate the efficiency and effectiveness of the proposed framework.

Dongwei Ye, Mengwu Guo

One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in poor estimations when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to several typical scenarios for dynamical systems: identification and estimation with an affine parametrization, nonlinear parametric approximation without prior knowledge, and general parameter estimation for a given dynamical system.

Marc Fransen, Andreas Fürst, Deepak Tunuguntla et al.

Micro-scale mechanisms, such as inter-particle and particle-fluid interactions, govern the behaviour of granular systems. While particle-scale simulations provide detailed insights into these interactions, their computational cost is often prohibitive. Attended by researchers from both the granular materials (GM) and machine learning (ML) communities, a recent Lorentz Center Workshop on "Machine Learning for Discrete Granular Media" brought the ML community up to date with GM challenges. This position paper emerged from the workshop discussions. We define granular materials and identify seven key challenges that characterise their distinctive behaviour across various scales and regimes, ranging from gas-like to fluid-like and solid-like. Addressing these challenges is essential for developing robust and efficient digital twins for granular systems in various industrial applications. To showcase the potential of ML to the GM community, we present classical and emerging machine/deep learning techniques that have been, or could be, applied to granular materials. We reviewed sequence-based learning models for path-dependent constitutive behaviour, followed by encoder-decoder type models for representing high-dimensional data. We then explore graph neural networks and recent advances in neural operator learning. Lastly, we discuss model-order reduction and probabilistic learning techniques for high-dimensional parameterised systems, which are crucial for quantifying uncertainties arising from physics-based and data-driven models. We present a workflow aimed at unifying data structures and modelling pipelines and guiding readers through the selection, training, and deployment of ML surrogates for granular material simulations. Finally, we illustrate the workflow's practical use with two representative examples, focusing on granular materials in solid-like and fluid-like regimes.

Dongwei Ye, Weihao Yan, Christoph Brune et al.

Gaussian process regression is widely applied in computational science and engineering for surrogate modeling owning to its kernel-based and probabilistic nature. In this work, we propose a Bayesian approach that integrates the variability of input data into the Gaussian process regression for function and partial differential equation approximation. Leveraging two types of observables -- noise-corrupted outputs with certain inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution of uncertain inputs is estimated via Bayesian inference. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The setting of two types of data aligned with common scenarios of constructing surrogate models for the solutions of partial differential equations, where the data of boundary conditions and initial conditions are typically known while the data of solution may involve uncertainties due to the measurement or stochasticity. The effectiveness of the proposed method is demonstrated through several numerical examples including multiple one-dimensional functions, the heat equation and Allen-Cahn equation. A consistently good performance of generalization is observed, and a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.

Dongwei Ye, Pavel Zun, Valeria Krzhizhanovskaya et al.

In-Stent Restenosis is a recurrence of coronary artery narrowing due to vascular injury caused by balloon dilation and stent placement. It may lead to the relapse of angina symptoms or to an acute coronary syndrome. An uncertainty quantification of a model for In-Stent Restenosis with four uncertain parameters (endothelium regeneration time, the threshold strain for smooth muscle cells bond breaking, blood flow velocity and the percentage of fenestration in the internal elastic lamina) is presented. Two quantities of interest were studied, namely the average cross-sectional area and the maximum relative area loss in a vessel. Due to the computational intensity of the model and the number of evaluations required in the uncertainty quantification, a surrogate model, based on Gaussian process regression with proper orthogonal decomposition, was developed which subsequently replaced the original In-Stent Restenosis model in the uncertainty quantification. A detailed analysis of the uncertainty propagation and sensitivity analysis is presented. Around 11% and 16% of uncertainty are observed on the average cross-sectional area and maximum relative area loss respectively, and the uncertainty estimates show that a higher fenestration mainly determines uncertainty in the neointimal growth at the initial stage of the process. On the other hand, the uncertainty in blood flow velocity and endothelium regeneration time mainly determine the uncertainty in the quantities of interest at the later, clinically relevant stages of the restenosis process. The uncertainty in the threshold strain is relatively small compared to the other uncertain parameters.