Christopher C. Pain

Sibo Cheng, Jianhua Chen, Charitos Anastasiou et al.

Reduced-order modelling and low-dimensional surrogate models generated using machine learning algorithms have been widely applied in high-dimensional dynamical systems to improve the algorithmic efficiency. In this paper, we develop a system which combines reduced-order surrogate models with a novel data assimilation (DA) technique used to incorporate real-time observations from different physical spaces. We make use of local smooth surrogate functions which link the space of encoded system variables and the one of current observations to perform variational DA with a low computational cost. The new system, named Generalised Latent Assimilation can benefit both the efficiency provided by the reduced-order modelling and the accuracy of data assimilation. A theoretical analysis of the difference between surrogate and original assimilation cost function is also provided in this paper where an upper bound, depending on the size of the local training set, is given. The new approach is tested on a high-dimensional CFD application of a two-phase liquid flow with non-linear observation operators that current Latent Assimilation methods can not handle. Numerical results demonstrate that the proposed assimilation approach can significantly improve the reconstruction and prediction accuracy of the deep learning surrogate model which is nearly 1000 times faster than the CFD simulation.

Jucai Zhai, Muhammad Abdullah, Boyang Chen et al.

In scientific computing, the formulation of numerical discretisations of partial differential equations (PDEs) as untrained convolutional layers within Convolutional Neural Networks (CNNs), referred to by some as Neural Physics, has demonstrated good efficiency for executing physics-based solvers on GPUs. However, classical grid-based methods still face computational bottlenecks when solving problems involving billions of degrees of freedom. To address this challenge, this paper proposes a novel framework called 'Quantum Neural Physics' and develops a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). This approach maps analytically-determined stencils of discretised differential operators into parameter-free or untrained quantum convolutional kernels. By leveraging amplitude encoding, the Linear Combination of Unitaries technique and the Quantum Fourier Transform, the resulting quantum convolutional operators can be implemented using quantum circuits with a circuit depth that scales as O(log K), where K denotes the size of the encoded input block. These quantum operators are embedded into a classical W-Cycle multigrid using a U-Net. This design enables seamless integration of quantum operators within a hierarchical solver whilst retaining the robustness and convergence properties of classical multigrid methods. The proposed Quantum Neural Physics solver is validated on a quantum simulator for the Poisson equation, diffusion equation, convection-diffusion equation and incompressible Navier-Stokes equations. The solutions of the HQC-CNNMG are in close agreement with those from traditional solution methods. This work establishes a mapping from discretised physical equations to logarithmic-scale quantum circuits, providing a new and exploratory path to exponential memory compression and computational acceleration for PDE solvers on future fault-tolerant quantum computers.

Nathalie C. Pinheiro, Donghu Guo, Hannah P. Menke et al.

Modelling rock-fluid interaction requires solving a set of partial differential equations (PDEs) to predict the flow behaviour and the reactions of the fluid with the rock on the interfaces. Conventional high-fidelity numerical models require a high resolution to obtain reliable results, resulting in huge computational expense. This restricts the applicability of these models for multi-query problems, such as uncertainty quantification and optimisation, which require running numerous scenarios. As a cheaper alternative to high-fidelity models, this work develops eight surrogate models for predicting the fluid flow in porous media. Four of these are reduced-order models (ROM) based on one neural network for compression and another for prediction. The other four are single neural networks with the property of grid-size invariance; a term which we use to refer to image-to-image models that are capable of inferring on computational domains that are larger than those used during training. In addition to the novel grid-size-invariant framework for surrogate models, we compare the predictive performance of UNet and UNet++ architectures, and demonstrate that UNet++ outperforms UNet for surrogate models. Furthermore, we show that the grid-size-invariant approach is a reliable way to reduce memory consumption during training, resulting in good correlation between predicted and ground-truth values and outperforming the ROMs analysed. The application analysed is particularly challenging because fluid-induced rock dissolution results in a non-static solid field and, consequently, it cannot be used to help in adjustments of the future prediction.

Boyang Chen, Claire E. Heaney, Jefferson L. M. A. Gomes et al.

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier-Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov-Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes.

Boyang Chen, Claire E. Heaney, Christopher C. Pain

Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to implement numerical discretisations as convolutional layers of a neural network, where the weights or filters are determined analytically rather than by training. Furthermore, we demonstrate that these systems can be solved entirely by functions in AI libraries, either by using Jacobi iteration or multigrid methods, the latter realised through a U-Net architecture. Some advantages of the Neural Physics approach are that (1) the methods are platform agnostic; (2) the resulting solvers are fully differentiable, ideal for optimisation tasks; and (3) writing CFD solvers as (untrained) neural networks means that they can be seamlessly integrated with trained neural networks to form hybrid models. We demonstrate the proposed approach on a number of test cases of increasing complexity from advection-diffusion problems, the non-linear Burgers equation to the Navier-Stokes equations. We validate the approach by comparing our results with solutions obtained from traditionally written code and common benchmarks from the literature. We show that the proposed methodology can solve all these problems using repurposed AI libraries in an efficient way, without training, and presents a new avenue to explore in the development of methods to solve PDEs with implicit methods.

Nazanin Zounemat Kermani, Sadjad Naderi, Claire H. Dilliway et al.

Air pollution poses a significant threat to public health, causing or exacerbating many respiratory and cardiovascular diseases. In addition, climate change is bringing about more extreme weather events such as wildfires and heatwaves, which can increase levels of pollution and worsen the effects of pollution exposure. Recent advances in personal sensing have transformed the collection of behavioural and physiological data, leading to the potential for new improvements in healthcare. We wish to capitalise on this data, alongside new capabilities in AI for making time series predictions, in order to monitor and predict health outcomes for an individual. Thus, we present a novel workflow for predicting personalised health responses to pollution by integrating physiological data from wearable fitness devices with real-time environmental exposures. The data is collected from various sources in a secure and ethical manner, and is used to train an AI model to predict individual health responses to pollution exposure within a cloud-based, modular framework. We demonstrate that the AI model -- an Adversarial Autoencoder neural network in this case -- accurately reconstructs time-dependent health signals and captures nonlinear responses to pollution. Transfer learning is applied using data from a personal smartwatch, which increases the generalisation abilities of the AI model and illustrates the adaptability of the approach to real-world, user-generated data.

Claire E. Heaney, Zef Wolffs, Jón Atli Tómasson et al.

The modelling of multiphase flow in a pipe presents a significant challenge for high-resolution computational fluid dynamics (CFD) models due to the high aspect ratio (length over diameter) of the domain. In subsea applications, the pipe length can be several hundreds of kilometres versus a pipe diameter of just a few inches. In this paper, we present a new AI-based non-intrusive reduced-order model within a domain decomposition framework (AI-DDNIROM) which is capable of making predictions for domains significantly larger than the domain used in training. This is achieved by using domain decomposition; dimensionality reduction; training a neural network to make predictions for a single subdomain; and by using an iteration-by-subdomain technique to converge the solution over the whole domain. To find the low-dimensional space, we explore several types of autoencoder networks, known for their ability to compress information accurately and compactly. The performance of the autoencoders is assessed on two advection-dominated problems: flow past a cylinder and slug flow in a pipe. To make predictions in time, we exploit an adversarial network which aims to learn the distribution of the training data, in addition to learning the mapping between particular inputs and outputs. This type of network has shown the potential to produce realistic outputs. The whole framework is applied to multiphase slug flow in a horizontal pipe for which an AI-DDNIROM is trained on high-fidelity CFD simulations of a pipe of length 10 m with an aspect ratio of 13:1, and tested by simulating the flow for a pipe of length 98 m with an aspect ratio of almost 130:1. Statistics of the flows obtained from the CFD simulations are compared to those of the AI-DDNIROM predictions to demonstrate the success of our approach.

Vinicius L. S. Silva, Claire E. Heaney, Nenko Nenov et al.

We propose a new method in which a generative network (GN) integrate into a reduced-order model (ROM) framework is used to solve inverse problems for partial differential equations (PDE). The aim is to match available measurements and estimate the corresponding uncertainties associated with the states and parameters of a numerical physical simulation. The GN is trained using only unconditional simulations of the discretized PDE model. We compare the proposed method with the golden standard Markov chain Monte Carlo. We apply the proposed approaches to a spatio-temporal compartmental model in epidemiology. The results show that the proposed GN-based ROM can efficiently quantify uncertainty and accurately match the measurements and the golden standard, using only a few unconditional simulations of the full-order numerical PDE model.

Vinicius L. S. Silva, Claire E. Heaney, Yaqi Li et al.

We propose the novel use of a generative adversarial network (GAN) (i) to make predictions in time (PredGAN) and (ii) to assimilate measurements (DA-PredGAN). In the latter case, we take advantage of the natural adjoint-like properties of generative models and the ability to simulate forwards and backwards in time. GANs have received much attention recently, after achieving excellent results for their generation of realistic-looking images. We wish to explore how this property translates to new applications in computational modelling and to exploit the adjoint-like properties for efficient data assimilation. To predict the spread of COVID-19 in an idealised town, we apply these methods to a compartmental model in epidemiology that is able to model space and time variations. To do this, the GAN is set within a reduced-order model (ROM), which uses a low-dimensional space for the spatial distribution of the simulation states. Then the GAN learns the evolution of the low-dimensional states over time. The results show that the proposed methods can accurately predict the evolution of the high-fidelity numerical simulation, and can efficiently assimilate observed data and determine the corresponding model parameters.

César Quilodrán-Casas, Vinicius Santos Silva, Rossella Arcucci et al.

The outbreak of the coronavirus disease 2019 (COVID-19) has now spread throughout the globe infecting over 150 million people and causing the death of over 3.2 million people. Thus, there is an urgent need to study the dynamics of epidemiological models to gain a better understanding of how such diseases spread. While epidemiological models can be computationally expensive, recent advances in machine learning techniques have given rise to neural networks with the ability to learn and predict complex dynamics at reduced computational costs. Here we introduce two digital twins of a SEIRS model applied to an idealised town. The SEIRS model has been modified to take account of spatial variation and, where possible, the model parameters are based on official virus spreading data from the UK. We compare predictions from a data-corrected Bidirectional Long Short-Term Memory network and a predictive Generative Adversarial Network. The predictions given by these two frameworks are accurate when compared to the original SEIRS model data. Additionally, these frameworks are data-agnostic and could be applied to towns, idealised or real, in the UK or in other countries. Also, more compartments could be included in the SEIRS model, in order to study more realistic epidemiological behaviour.

Claire E. Heaney, Yuling Li, Omar K. Matar et al.

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to the approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE) although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with an SFC. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.

Toby Phillips, Claire E. Heaney, Paul N. Smith et al.

Using an autoencoder for dimensionality reduction, this paper presents a novel projection-based reduced-order model for eigenvalue problems. Reduced-order modelling relies on finding suitable basis functions which define a low-dimensional space in which a high-dimensional system is approximated. Proper orthogonal decomposition (POD) and singular value decomposition (SVD) are often used for this purpose and yield an optimal linear subspace. Autoencoders provide a nonlinear alternative to POD/SVD, that may capture, more efficiently, features or patterns in the high-fidelity model results. Reduced-order models based on an autoencoder and a novel hybrid SVD-autoencoder are developed. These methods are compared with the standard POD-Galerkin approach and are applied to two test cases taken from the field of nuclear reactor physics.