arXiv subjects

George Em Karniadakis

Publications and source records attributed to George Em Karniadakis.

At least 145 records · Page 8Linked to original sources

arXiv2023

A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data

We propose a framework and an algorithm to uncover the unknown parts of nonlinear equations directly from data. The framework is based on eXtended Physics-Informed Neural Networks (X-PINNs), domain decomposition in space-time, but we augment the original X-PINN method by imposing flux continuity across the domain interfaces. The well-known Allen-Cahn equation is used to demonstrate the approach. The Frobenius matrix norm is used to evaluate the accuracy of the X-PINN predictions and the results show excellent performance. In addition, symbolic regression is employed to determine the closed form of the unknown part of the equation from the data, and the results confirm the accuracy of the X-PINNs based approach. To test the framework in a situation resembling real-world data, random noise is added to the datasets to mimic scenarios such as the presence of thermal noise or instrument errors. The results show that the framework is stable against significant amount of noise. As the final part, we determine the minimal amount of data required for training the neural network. The framework is able to predict the correct form and coefficients of the underlying dynamical equation when at least 50\% data is used for training.

cond-mat.dis-nn↗

arXiv2023

A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimes

Quantifying biomechanical properties of the human vasculature could deepen our understanding of cardiovascular diseases. Standard nonlinear regression in constitutive modeling requires considerable high-quality data and an explicit form of the constitutive model as prior knowledge. By contrast, we propose a novel approach that combines generative deep learning with Bayesian inference to efficiently infer families of constitutive relationships in data-sparse regimes. Inspired by the concept of functional priors, we develop a generative adversarial network (GAN) that incorporates a neural operator as the generator and a fully-connected neural network as the discriminator. The generator takes a vector of noise conditioned on measurement data as input and yields the predicted constitutive relationship, which is scrutinized by the discriminator in the following step. We demonstrate that this framework can accurately estimate means and standard deviations of the constitutive relationships of the murine aorta using data collected either from model-generated synthetic data or ex vivo experiments for mice with genetic deficiencies. In addition, the framework learns priors of constitutive models without explicitly knowing their functional form, providing a new model-agnostic approach to learning hidden constitutive behaviors from data.

cs.LG↗

arXiv2023

Learning in latent spaces improves the predictive accuracy of deep neural operators

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate mappings between infinite-dimensional Banach spaces. As data-driven models, neural operators require the generation of labeled observations, which in cases of complex high-fidelity models result in high-dimensional datasets containing redundant and noisy features, which can hinder gradient-based optimization. Mapping these high-dimensional datasets to a low-dimensional latent space of salient features can make it easier to work with the data and also enhance learning. In this work, we investigate the latent deep operator network (L-DeepONet), an extension of standard DeepONet, which leverages latent representations of high-dimensional PDE input and output functions identified with suitable autoencoders. We illustrate that L-DeepONet outperforms the standard approach in terms of both accuracy and computational efficiency across diverse time-dependent PDEs, e.g., modeling the growth of fracture in brittle materials, convective fluid flows, and large-scale atmospheric flows exhibiting multiscale dynamical features.

cs.LG↗

arXiv2023

Bi-orthogonal fPINN: A physics-informed neural network method for solving time-dependent stochastic fractional PDEs

Fractional partial differential equations (FPDEs) can effectively represent anomalous transport and nonlocal interactions. However, inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties. Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations (SFPDEs). There are many challenges in solving SFPDEs numerically, especially for long-time integration since such problems are high-dimensional and nonlocal. Here, we combine the bi-orthogonal (BO) method for representing stochastic processes with physics-informed neural networks (PINNs) for solving partial differential equations to formulate the bi-orthogonal PINN method (BO-fPINN) for solving time-dependent SFPDEs. Specifically, we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs, and include the BO constraints in the loss function following a weak formulation. Since automatic differentiation is not currently applicable to fractional derivatives, we employ discretization on a grid to to compute the fractional derivatives of the neural network output. The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings. Moreover, the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems. We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems. The results demonstrate the flexibility and efficiency of the proposed method, especially for inverse problems.

math.NA↗

arXiv2023

ViTO: Vision Transformer-Operator

We combine vision transformers with operator learning to solve diverse inverse problems described by partial differential equations (PDEs). Our approach, named ViTO, combines a U-Net based architecture with a vision transformer. We apply ViTO to solve inverse PDE problems of increasing complexity, namely for the wave equation, the Navier-Stokes equations and the Darcy equation. We focus on the more challenging case of super-resolution, where the input dataset for the inverse problem is at a significantly coarser resolution than the output. The results we obtain are comparable or exceed the leading operator network benchmarks in terms of accuracy. Furthermore, ViTO`s architecture has a small number of trainable parameters (less than 10% of the leading competitor), resulting in a performance speed-up of over 5x when averaged over the various test cases.

cs.CV↗

arXiv2023

Dynamic spreading and infiltration of a molten sand droplet on a porous surface

Compared to smooth surfaces, droplet spreading on porous surfaces is more complex and has relevance in many engineering applications. In this work, we investigate the infiltration dynamics of molten sand droplets on structured porous surfaces using the multiphase many-body dissipative particle dynamics (mDPD) method. We carry out three-dimensional simulations with different equilibrium contact angles and surface porosities. The temporal evolution of the radius of the wetted area follows a power law, as in the case of a smooth surface. The infiltration rate on the other hand is dictated by the competition between spreading and capillary inhibition of the pores. Additionally, the temporal evolution of the droplet height and the contact angle on the porous surface is also presented.

physics.flu-dyn↗

arXiv2023

Learning stiff chemical kinetics using extended deep neural operators

We utilize neural operators to learn the solution propagator for the challenging chemical kinetics equation. Specifically, we apply the deep operator network (DeepONet) along with its extensions, such as the autoencoder-based DeepONet and the newly proposed Partition-of-Unity (PoU-) DeepONet to study a range of examples, including the ROBERS problem with three species, the POLLU problem with 25 species, pure kinetics of the syngas skeletal model for $CO/H_2$ burning, which contains 11 species and 21 reactions and finally, a temporally developing planar $CO/H_2$ jet flame (turbulent flame) using the same syngas mechanism. We have demonstrated the advantages of the proposed approach through these numerical examples. Specifically, to train the DeepONet for the syngas model, we solve the skeletal kinetic model for different initial conditions. In the first case, we parametrize the initial conditions based on equivalence ratios and initial temperature values. In the second case, we perform a direct numerical simulation of a two-dimensional temporally developing $CO/H_2$ jet flame. Then, we initialize the kinetic model by the thermochemical states visited by a subset of grid points at different time snapshots. Stiff problems are computationally expensive to solve with traditional stiff solvers. Thus, this work aims to develop a neural operator-based surrogate model to solve stiff chemical kinetics. The operator, once trained offline, can accurately integrate the thermochemical state for arbitrarily large time advancements, leading to significant computational gains compared to stiff integration schemes.

physics.chem-ph↗

arXiv2023

Learning bias corrections for climate models using deep neural operators

Numerical simulation for climate modeling resolving all important scales is a computationally taxing process. Therefore, to circumvent this issue a low resolution simulation is performed, which is subsequently corrected for bias using reanalyzed data (ERA5), known as nudging correction. The existing implementation for nudging correction uses a relaxation based method for the algebraic difference between low resolution and ERA5 data. In this study, we replace the bias correction process with a surrogate model based on the Deep Operator Network (DeepONet). DeepONet (Deep Operator Neural Network) learns the mapping from the state before nudging (a functional) to the nudging tendency (another functional). The nudging tendency is a very high dimensional data albeit having many low energy modes. Therefore, the DeepoNet is combined with a convolution based auto-encoder-decoder (AED) architecture in order to learn the nudging tendency in a lower dimensional latent space efficiently. The accuracy of the DeepONet model is tested against the nudging tendency obtained from the E3SMv2 (Energy Exascale Earth System Model) and shows good agreement. The overarching goal of this work is to deploy the DeepONet model in an online setting and replace the nudging module in the E3SM loop for better efficiency and accuracy.

physics.ao-ph↗

arXiv2023

Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils

Deep neural operators, such as DeepONets, have changed the paradigm in high-dimensional nonlinear regression from function regression to (differential) operator regression, paving the way for significant changes in computational engineering applications. Here, we investigate the use of DeepONets to infer flow fields around unseen airfoils with the aim of shape optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results which display little to no degradation in prediction accuracy, while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as their shape can be easily defined by the four-digit parametrization. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work.

physics.flu-dyn↗

arXiv2023

A Hybrid Deep Neural Operator/Finite Element Method for Ice-Sheet Modeling

One of the most challenging and consequential problems in climate modeling is to provide probabilistic projections of sea level rise. A large part of the uncertainty of sea level projections is due to uncertainty in ice sheet dynamics. At the moment, accurate quantification of the uncertainty is hindered by the cost of ice sheet computational models. In this work, we develop a hybrid approach to approximate existing ice sheet computational models at a fraction of their cost. Our approach consists of replacing the finite element model for the momentum equations for the ice velocity, the most expensive part of an ice sheet model, with a Deep Operator Network, while retaining a classic finite element discretization for the evolution of the ice thickness. We show that the resulting hybrid model is very accurate and it is an order of magnitude faster than the traditional finite element model. Further, a distinctive feature of the proposed model compared to other neural network approaches, is that it can handle high-dimensional parameter spaces (parameter fields) such as the basal friction at the bed of the glacier, and can therefore be used for generating samples for uncertainty quantification. We study the impact of hyper-parameters, number of unknowns and correlation length of the parameter distribution on the training and accuracy of the Deep Operator Network on a synthetic ice sheet model. We then target the evolution of the Humboldt glacier in Greenland and show that our hybrid model can provide accurate statistics of the glacier mass loss and can be effectively used to accelerate the quantification of uncertainty.

physics.comp-ph↗

arXiv2023

L-HYDRA: Multi-Head Physics-Informed Neural Networks

We introduce multi-head neural networks (MH-NNs) to physics-informed machine learning, which is a type of neural networks (NNs) with all nonlinear hidden layers as the body and multiple linear output layers as multi-head. Hence, we construct multi-head physics-informed neural networks (MH-PINNs) as a potent tool for multi-task learning (MTL), generative modeling, and few-shot learning for diverse problems in scientific machine learning (SciML). MH-PINNs connect multiple functions/tasks via a shared body as the basis functions as well as a shared distribution for the head. The former is accomplished by solving multiple tasks with MH-PINNs with each head independently corresponding to each task, while the latter by employing normalizing flows (NFs) for density estimate and generative modeling. To this end, our method is a two-stage method, and both stages can be tackled with standard deep learning tools of NNs, enabling easy implementation in practice. MH-PINNs can be used for various purposes, such as approximating stochastic processes, solving multiple tasks synergistically, providing informative prior knowledge for downstream few-shot learning tasks such as meta-learning and transfer learning, learning representative basis functions, and uncertainty quantification. We demonstrate the effectiveness of MH-PINNs in five benchmarks, investigating also the possibility of synergistic learning in regression analysis. We name the open-source code "Lernaean Hydra" (L-HYDRA), since this mythical creature possessed many heads for performing important multiple tasks, as in the proposed method.

cs.LG↗

arXiv2022

How important are activation functions in regression and classification? A survey, performance comparison, and future directions

Inspired by biological neurons, the activation functions play an essential part in the learning process of any artificial neural network commonly used in many real-world problems. Various activation functions have been proposed in the literature for classification as well as regression tasks. In this work, we survey the activation functions that have been employed in the past as well as the current state-of-the-art. In particular, we present various developments in activation functions over the years and the advantages as well as disadvantages or limitations of these activation functions. We also discuss classical (fixed) activation functions, including rectifier units, and adaptive activation functions. In addition to discussing the taxonomy of activation functions based on characterization, a taxonomy of activation functions based on applications is presented. To this end, the systematic comparison of various fixed and adaptive activation functions is performed for classification data sets such as the MNIST, CIFAR-10, and CIFAR- 100. In recent years, a physics-informed machine learning framework has emerged for solving problems related to scientific computations. For this purpose, we also discuss various requirements for activation functions that have been used in the physics-informed machine learning framework. Furthermore, various comparisons are made among different fixed and adaptive activation functions using various machine learning libraries such as TensorFlow, Pytorch, and JAX.

cs.LG↗

arXiv2022

Reliable extrapolation of deep neural operators informed by physics or sparse observations

Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. Pure data-driven neural operators and deep learning models, in general, are usually limited to interpolation scenarios, where new predictions utilize inputs within the support of the training set. However, in the inference stage of real-world applications, the input may lie outside the support, i.e., extrapolation is required, which may result to large errors and unavoidable failure of deep learning models. Here, we address this challenge of extrapolation for deep neural operators. First, we systematically investigate the extrapolation behavior of DeepONets by quantifying the extrapolation complexity via the 2-Wasserstein distance between two function spaces and propose a new behavior of bias-variance trade-off for extrapolation with respect to model capacity. Subsequently, we develop a complete workflow, including extrapolation determination, and we propose five reliable learning methods that guarantee a safe prediction under extrapolation by requiring additional information -- the governing PDEs of the system or sparse new observations. The proposed methods are based on either fine-tuning a pre-trained DeepONet or multifidelity learning. We demonstrate the effectiveness of the proposed framework for various types of parametric PDEs. Our systematic comparisons provide practical guidelines for selecting a proper extrapolation method depending on the available information, desired accuracy, and required inference speed.

cs.LG↗

arXiv2022

Deep transfer operator learning for partial differential equations under conditional shift

Transfer learning (TL) enables the transfer of knowledge gained in learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labeling, potential computational power limitations, and dataset distribution mismatches. We propose a new TL framework for task-specific learning (functional regression in partial differential equations (PDEs)) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of target data. Inspired by the conditional embedding operator theory, we minimize the statistical distance between labeled target data and the surrogate prediction on unlabeled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various TL scenarios involving nonlinear PDEs under diverse conditions due to shift in the geometric domain and model dynamics. Our TL framework enables fast and efficient learning of heterogeneous tasks despite significant differences between the source and target domains.

cs.LG↗

arXiv2022

SMS: Spiking Marching Scheme for Efficient Long Time Integration of Differential Equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

math.NA↗

arXiv2022

When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?

Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, Extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding on their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multi-layer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem, and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.

cs.LG↗

arXiv2022

Spiking Neural Operators for Scientific Machine Learning

The main computational task of Scientific Machine Learning (SciML) is function regression, required both for inputs as well as outputs of a simulation. Physics-Informed Neural Networks (PINNs) and neural operators (such as DeepONet) have been very effective in solving Partial Differential Equations (PDEs), but they tax computational resources heavily and cannot be readily adopted for edge computing. Here, we address this issue by considering Spiking Neural Networks (SNNs), which have shown promise in reducing energy consumption by two orders of magnitude or more. We present a SNN-based method to perform regression, which has been a challenge due to the inherent difficulty in representing a function's input domain and continuous output values as spikes. We first propose a new method for encoding continuous values into spikes based on a triangular matrix in space and time, and demonstrate its better performance compared to the existing methods. Next, we demonstrate that using a simple SNN architecture consisting of Leaky Integrate and Fire (LIF) activation and two dense layers, we can achieve relatively accurate function regression results. Moreover, we can replace the LIF with a trained Multi-Layer Perceptron (MLP) network and obtain comparable results but three times faster. Then, we introduce the DeepONet, consisting of a branch (typically a Fully-connected Neural Network, FNN) for inputs and a trunk (also a FNN) for outputs. We can build a spiking DeepONet by either replacing the branch or the trunk by a SNN. We demonstrate this new approach for classification using the SNN in the branch, achieving results comparable to the literature. Finally, we design a spiking DeepONet for regression by replacing its trunk with a SNN, and achieve good accuracy for approximating functions as well as inferring solutions of differential equations.

cs.NE↗

arXiv2022

Discovering and forecasting extreme events via active learning in neural operators

Extreme events in society and nature, such as pandemic spikes, rogue waves, or structural failures, can have catastrophic consequences. Characterizing extremes is difficult as they occur rarely, arise from seemingly benign conditions, and belong to complex and often unknown infinite-dimensional systems. Such challenges render attempts at characterizing them as moot. We address each of these difficulties by combining novel training schemes in Bayesian experimental design (BED) with an ensemble of deep neural operators (DNOs). This model-agnostic framework pairs a BED scheme that actively selects data for quantifying extreme events with an ensemble of DNOs that approximate infinite-dimensional nonlinear operators. We find that not only does this framework clearly beat Gaussian processes (GPs) but that 1) shallow ensembles of just two members perform best; 2) extremes are uncovered regardless of the state of initial data (i.e. with or without extremes); 3) our method eliminates "double-descent" phenomena; 4) the use of batches of suboptimal acquisition points compared to step-by-step global optima does not hinder BED performance; and 5) Monte Carlo acquisition outperforms standard optimizers in high-dimensions. Together these conclusions form the foundation of an AI-assisted experimental infrastructure that can efficiently infer and pinpoint critical situations across many domains, from physical to societal systems.

cs.LG↗