244 research outputs found
Non-linear coupled CNN models for multiscale image analysis
A CNN model of partial differential equations (PDEs) for image multiscale analysis is proposed. The model is based on a polynomial representation of the diffusivity function and defines a paradigm of polynomial CNNs,for approximating a large class of nonlinear isotropic and/or anisotropic PDEs. The global dynamics of spacediscrete polynomial CNN models is analyzed and compared with the dynamic behavior of the corresponding space-continuous PDE models. It is shown that in the isotropic case the two models are not topologically equivalent: in particular discrete CNN models allow one to obtain the output image without stopping the image evolution after a given time (scale). This property represents an advantage with respect to continuous PDE models and could simplify some image preprocessing algorithm
Generalised cellular neural networks (GCNNs) constructed using particle swarm optimisation for spatio-temporal evolutionary pattern identification
Particle swarm optimization (PSO) is introduced to implement a new constructive learning algorithm for training generalized cellular neural networks (GCNNs) for the identification of spatio-temporal evolutionary (STE) systems. The basic idea of the new PSO-based learning algorithm is to successively approximate the desired signal by progressively pursuing relevant orthogonal projections. This new algorithm will thus be referred to as the orthogonal projection pursuit (OPP) algorithm, which is in mechanism similar to the conventional projection pursuit approach. A novel two-stage hybrid training scheme is proposed for constructing a parsimonious GCNN model. In the first stage, the orthogonal projection pursuit algorithm is applied to adaptively and successively augment the network, where adjustable parameters of the associated units are optimized using a particle swarm optimizer. The resultant network model produced at the first stage may be redundant. In the second stage, a forward orthogonal regression (FOR) algorithm, aided by mutual information estimation, is applied to re. ne and improve the initially trained network. The effectiveness and performance of the proposed method is validated by applying the new modeling framework to a spatio-temporal evolutionary system identification problem
Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks
As an alternative to classical numerical solvers for partial differential
equations (PDEs) subject to boundary value constraints, there has been a surge
of interest in investigating neural networks that can solve such problems
efficiently. In this work, we design a general solution operator for two
different time-independent PDEs using graph neural networks (GNNs) and spectral
graph convolutions. We train the networks on simulated data from a finite
elements solver on a variety of shapes and inhomogeneities. In contrast to
previous works, we focus on the ability of the trained operator to generalize
to previously unseen scenarios. Specifically, we test generalization to meshes
with different shapes and superposition of solutions for a different number of
inhomogeneities. We find that training on a diverse dataset with lots of
variation in the finite element meshes is a key ingredient for achieving good
generalization results in all cases. With this, we believe that GNNs can be
used to learn solution operators that generalize over a range of properties and
produce solutions much faster than a generic solver. Our dataset, which we make
publicly available, can be used and extended to verify the robustness of these
models under varying conditions
Multipole Graph Neural Operator for Parametric Partial Differential Equations
One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time
Self-Supervised Learning with Lie Symmetries for Partial Differential Equations
Machine learning for differential equations paves the way for computationally
efficient alternatives to numerical solvers, with potentially broad impacts in
science and engineering. Though current algorithms typically require simulated
training data tailored to a given setting, one may instead wish to learn useful
information from heterogeneous sources, or from real dynamical systems
observations that are messy or incomplete. In this work, we learn
general-purpose representations of PDEs from heterogeneous data by implementing
joint embedding methods for self-supervised learning (SSL), a framework for
unsupervised representation learning that has had notable success in computer
vision. Our representation outperforms baseline approaches to invariant tasks,
such as regressing the coefficients of a PDE, while also improving the
time-stepping performance of neural solvers. We hope that our proposed
methodology will prove useful in the eventual development of general-purpose
foundation models for PDEs
PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers
Time-dependent partial differential equations (PDEs) are ubiquitous in
science and engineering. Recently, mostly due to the high computational cost of
traditional solution techniques, deep neural network based surrogates have
gained increased interest. The practical utility of such neural PDE solvers
relies on their ability to provide accurate, stable predictions over long time
horizons, which is a notoriously hard problem. In this work, we present a
large-scale analysis of common temporal rollout strategies, identifying the
neglect of non-dominant spatial frequency information, often associated with
high frequencies in PDE solutions, as the primary pitfall limiting stable,
accurate rollout performance. Based on these insights, we draw inspiration from
recent advances in diffusion models to introduce PDE-Refiner; a novel model
class that enables more accurate modeling of all frequency components via a
multistep refinement process. We validate PDE-Refiner on challenging benchmarks
of complex fluid dynamics, demonstrating stable and accurate rollouts that
consistently outperform state-of-the-art models, including neural, numerical,
and hybrid neural-numerical architectures. We further demonstrate that
PDE-Refiner greatly enhances data efficiency, since the denoising objective
implicitly induces a novel form of spectral data augmentation. Finally,
PDE-Refiner's connection to diffusion models enables an accurate and efficient
assessment of the model's predictive uncertainty, allowing us to estimate when
the surrogate becomes inaccurate.Comment: Project website: https://phlippe.github.io/PDERefiner
Frame invariant neural network closures for Kraichnan turbulence
Numerical simulations of geophysical and atmospheric flows have to rely on
parameterizations of subgrid scale processes due to their limited spatial
resolution. Despite substantial progress in developing parameterization (or
closure) models for subgrid scale (SGS) processes using physical insights and
mathematical approximations, they remain imperfect and can lead to inaccurate
predictions. In recent years, machine learning has been successful in
extracting complex patterns from high-resolution spatio-temporal data, leading
to improved parameterization models, and ultimately better coarse grid
prediction. However, the inability to satisfy known physics and poor
generalization hinders the application of these models for real-world problems.
In this work, we propose a frame invariant closure approach to improve the
accuracy and generalizability of deep learning-based subgrid scale closure
models by embedding physical symmetries directly into the structure of the
neural network. Specifically, we utilized specialized layers within the
convolutional neural network in such a way that desired constraints are
theoretically guaranteed without the need for any regularization terms. We
demonstrate our framework for a two-dimensional decaying turbulence test case
mostly characterized by the forward enstrophy cascade. We show that our frame
invariant SGS model (i) accurately predicts the subgrid scale source term, (ii)
respects the physical symmetries such as translation, Galilean, and rotation
invariance, and (iii) is numerically stable when implemented in coarse-grid
simulation with generalization to different initial conditions and Reynolds
number. This work builds a bridge between extensive physics-based theories and
data-driven modeling paradigms, and thus represents a promising step towards
the development of physically consistent data-driven turbulence closure models
Physics-Informed Computer Vision: A Review and Perspectives
Incorporation of physical information in machine learning frameworks are
opening and transforming many application domains. Here the learning process is
augmented through the induction of fundamental knowledge and governing physical
laws. In this work we explore their utility for computer vision tasks in
interpreting and understanding visual data. We present a systematic literature
review of formulation and approaches to computer vision tasks guided by
physical laws. We begin by decomposing the popular computer vision pipeline
into a taxonomy of stages and investigate approaches to incorporate governing
physical equations in each stage. Existing approaches in each task are analyzed
with regard to what governing physical processes are modeled, formulated and
how they are incorporated, i.e. modify data (observation bias), modify networks
(inductive bias), and modify losses (learning bias). The taxonomy offers a
unified view of the application of the physics-informed capability,
highlighting where physics-informed learning has been conducted and where the
gaps and opportunities are. Finally, we highlight open problems and challenges
to inform future research. While still in its early days, the study of
physics-informed computer vision has the promise to develop better computer
vision models that can improve physical plausibility, accuracy, data efficiency
and generalization in increasingly realistic applications
Data-Driven Exploration of Coarse-Grained Equations: Harnessing Machine Learning
In scientific research, understanding and modeling physical systems often involves working with complex equations called Partial Differential Equations (PDEs). These equations are essential for describing the relationships between variables and their derivatives, allowing us to analyze a wide range of phenomena, from fluid dynamics to quantum mechanics. Traditionally, the discovery of PDEs relied on mathematical derivations and expert knowledge. However, the advent of data-driven approaches and machine learning (ML) techniques has transformed this process. By harnessing ML techniques and data analysis methods, data-driven approaches have revolutionized the task of uncovering complex equations that describe physical systems. The primary goal in this thesis is to develop methodologies that can automatically extract simplified equations by training models using available data. ML algorithms have the ability to learn underlying patterns and relationships within the data, making it possible to extract simplified equations that capture the essential behavior of the system. This study considers three distinct learning categories: black-box, gray-box, and white-box learning.
The initial phase of the research focuses on black-box learning, where no prior information about the equations is available. Three different neural network architectures are explored: multi-layer perceptron (MLP), convolutional neural network (CNN), and a hybrid architecture combining CNN and long short-term memory (CNN-LSTM). These neural networks are applied to uncover the non-linear equations of motion associated with phase-field models, which include both non-conserved and conserved order parameters.
The second architecture explored in this study addresses explicit equation discovery in gray-box learning scenarios, where a portion of the equation is unknown. The framework employs eXtended Physics-Informed Neural Networks (X-PINNs) and incorporates domain decomposition in space to uncover a segment of the widely-known Allen-Cahn equation. Specifically, the Laplacian part of the equation is assumed to be known, while the objective is to discover the non-linear component of the equation. Moreover, symbolic regression techniques are applied to deduce the precise mathematical expression for the unknown segment of the equation.
Furthermore, the final part of the thesis focuses on white-box learning, aiming to uncover equations that offer a detailed understanding of the studied system. Specifically, a coarse parametric ordinary differential equation (ODE) is introduced to accurately capture the spreading radius behavior of Calcium-magnesium-aluminosilicate (CMAS) droplets. Through the utilization of the Physics-Informed Neural Network (PINN) framework, the parameters of this ODE are determined, facilitating precise estimation. The architecture is employed to discover the unknown parameters of the equation, assuming that all terms of the ODE are known. This approach significantly improves our comprehension of the spreading dynamics associated with CMAS droplets
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