arXiv subjects

Robert M. Kirby

Publications and source records attributed to Robert M. Kirby.

At least 37 records · Page 2Linked to original sources

arXiv2023

A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) - and, in fact, machine learning (ML) in general. This paper is meant to move towards addressing the latter through the study of PINNs on new tasks, for which parameterized PDEs provides a good testbed application as tasks can be easily defined in this context. Following the ML world, we introduce metalearning of PINNs with application to parameterized PDEs. By introducing metalearning and transfer learning concepts, we can greatly accelerate the PINNs optimization process. We present a survey of model-agnostic metalearning, and then discuss our model-aware metalearning applied to PINNs as well as implementation considerations and algorithmic complexity. We then test our approach on various canonical forward parameterized PDEs that have been presented in the emerging PINNs literature.

physics.comp-ph↗

arXiv2023

Weight Matrix Dimensionality Reduction in Deep Learning via Kronecker Multi-layer Architectures

Deep learning using neural networks is an effective technique for generating models of complex data. However, training such models can be expensive when networks have large model capacity resulting from a large number of layers and nodes. For training in such a computationally prohibitive regime, dimensionality reduction techniques ease the computational burden, and allow implementations of more robust networks. We propose a novel type of such dimensionality reduction via a new deep learning architecture based on fast matrix multiplication of a Kronecker product decomposition; in particular our network construction can be viewed as a Kronecker product-induced sparsification of an "extended" fully connected network. Analysis and practical examples show that this architecture allows a neural network to be trained and implemented with a significant reduction in computational time and resources, while achieving a similar error level compared to a traditional feedforward neural network.

cs.LG↗

arXiv2023

Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)

Multifidelity simulation methodologies are often used in an attempt to judiciously combine low-fidelity and high-fidelity simulation results in an accuracy-increasing, cost-saving way. Candidates for this approach are simulation methodologies for which there are fidelity differences connected with significant computational cost differences. Physics-informed Neural Networks (PINNs) are candidates for these types of approaches due to the significant difference in training times required when different fidelities (expressed in terms of architecture width and depth as well as optimization criteria) are employed. In this paper, we propose a particular multifidelity approach applied to PINNs that exploits low-rank structure. We demonstrate that width, depth, and optimization criteria can be used as parameters related to model fidelity, and show numerical justification of cost differences in training due to fidelity parameter choices. We test our multifidelity scheme on various canonical forward PDE models that have been presented in the emerging PINNs literature.

physics.comp-ph↗

arXiv2022

Infinite-Fidelity Coregionalization for Physical Simulation

Multi-fidelity modeling and learning are important in physical simulation-related applications. It can leverage both low-fidelity and high-fidelity examples for training so as to reduce the cost of data generation while still achieving good performance. While existing approaches only model finite, discrete fidelities, in practice, the fidelity choice is often continuous and infinite, which can correspond to a continuous mesh spacing or finite element length. In this paper, we propose Infinite Fidelity Coregionalization (IFC). Given the data, our method can extract and exploit rich information within continuous, infinite fidelities to bolster the prediction accuracy. Our model can interpolate and/or extrapolate the predictions to novel fidelities, which can be even higher than the fidelities of training data. Specifically, we introduce a low-dimensional latent output as a continuous function of the fidelity and input, and multiple it with a basis matrix to predict high-dimensional solution outputs. We model the latent output as a neural Ordinary Differential Equation (ODE) to capture the complex relationships within and integrate information throughout the continuous fidelities. We then use Gaussian processes or another ODE to estimate the fidelity-varying bases. For efficient inference, we reorganize the bases as a tensor, and use a tensor-Gaussian variational posterior to develop a scalable inference algorithm for massive outputs. We show the advantage of our method in several benchmark tasks in computational physics.

cs.LG↗

arXiv2022

Batch Multi-Fidelity Active Learning with Budget Constraints

Learning functions with high-dimensional outputs is critical in many applications, such as physical simulation and engineering design. However, collecting training examples for these applications is often costly, e.g. by running numerical solvers. The recent work (Li et al., 2022) proposes the first multi-fidelity active learning approach for high-dimensional outputs, which can acquire examples at different fidelities to reduce the cost while improving the learning performance. However, this method only queries at one pair of fidelity and input at a time, and hence has a risk to bring in strongly correlated examples to reduce the learning efficiency. In this paper, we propose Batch Multi-Fidelity Active Learning with Budget Constraints (BMFAL-BC), which can promote the diversity of training examples to improve the benefit-cost ratio, while respecting a given budget constraint for batch queries. Hence, our method can be more practically useful. Specifically, we propose a novel batch acquisition function that measures the mutual information between a batch of multi-fidelity queries and the target function, so as to penalize highly correlated queries and encourages diversity. The optimization of the batch acquisition function is challenging in that it involves a combinatorial search over many fidelities while subject to the budget constraint. To address this challenge, we develop a weighted greedy algorithm that can sequentially identify each (fidelity, input) pair, while achieving a near $(1 - 1/e)$-approximation of the optimum. We show the advantage of our method in several computational physics and engineering applications.

cs.LG↗

arXiv2022

ENO-Based High-Order Data-Bounded and Constrained Positivity-Preserving Interpolation

A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially non-oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which the physics is calculated (and back). Interpolating density or other key physical quantities without accounting for property preservation may lead to negative values that are nonphysical and result in inaccurate representations and/or interpretations of the physical data. Property-preserving interpolation is straightforward when used in the context of low-order numerical simulation methods. High-order property-preserving interpolation is, however, nontrivial, especially in the case where the interpolation points are not equispaced. In this paper, we demonstrate that it is possible to construct high-order interpolation methods that ensure either data boundedness or constrained positivity preservation. A novel feature of the algorithm is that the positivity-preserving interpolant is constrained; that is, the amount by which it exceeds the data values may be strictly controlled. The algorithm we have developed comes with theoretical estimates that provide sufficient conditions for data boundedness and constrained positivity preservation. We demonstrate the application of our algorithm on a collection of 1D and 2D numerical examples, and show that in all cases property preservation is respected.

math.NA↗

arXiv2022

Momentum Transformer: Closing the Performance Gap Between Self-attention and Its Linearization

Transformers have achieved remarkable success in sequence modeling and beyond but suffer from quadratic computational and memory complexities with respect to the length of the input sequence. Leveraging techniques include sparse and linear attention and hashing tricks; efficient transformers have been proposed to reduce the quadratic complexity of transformers but significantly degrade the accuracy. In response, we first interpret the linear attention and residual connections in computing the attention map as gradient descent steps. We then introduce momentum into these components and propose the \emph{momentum transformer}, which utilizes momentum to improve the accuracy of linear transformers while maintaining linear memory and computational complexities. Furthermore, we develop an adaptive strategy to compute the momentum value for our model based on the optimal momentum for quadratic optimization. This adaptive momentum eliminates the need to search for the optimal momentum value and further enhances the performance of the momentum transformer. A range of experiments on both autoregressive and non-autoregressive tasks, including image generation and machine translation, demonstrate that the momentum transformer outperforms popular linear transformers in training efficiency and accuracy.

cs.LG↗

arXiv2022

Adaptive Self-supervision Algorithms for Physics-informed Neural Networks

Physics-informed neural networks (PINNs) incorporate physical knowledge from the problem domain as a soft constraint on the loss function, but recent work has shown that this can lead to optimization difficulties. Here, we study the impact of the location of the collocation points on the trainability of these models. We find that the vanilla PINN performance can be significantly boosted by adapting the location of the collocation points as training proceeds. Specifically, we propose a novel adaptive collocation scheme which progressively allocates more collocation points (without increasing their number) to areas where the model is making higher errors (based on the gradient of the loss function in the domain). This, coupled with a judicious restarting of the training during any optimization stalls (by simply resampling the collocation points in order to adjust the loss landscape) leads to better estimates for the prediction error. We present results for several problems, including a 2D Poisson and diffusion-advection system with different forcing functions. We find that training vanilla PINNs for these problems can result in up to 70% prediction error in the solution, especially in the regime of low collocation points. In contrast, our adaptive schemes can achieve up to an order of magnitude smaller error, with similar computational complexity as the baseline. Furthermore, we find that the adaptive methods consistently perform on-par or slightly better than vanilla PINN method, even for large collocation point regimes. The code for all the experiments has been open sourced.

cs.LG↗

arXiv2022

Numerical Testing of a New Positivity-Preserving Interpolation Algorithm

An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application for this work was a numerical weather prediction (NWP) code that uses a spectral element mesh discretization for its dynamics core and a cartesian tensor product mesh for the evaluation of its physics routines. This coupling of spectral element mesh, which uses nonuniformly spaced quadrature/collocation points, and uniformly-spaced cartesian mesh combined with the desire to maintain positivity when moving between these meshes necessitates our work. This new approach is evaluated against several typical algorithms in use on a range of test problems in one or more space dimensions. The results obtained show that the new method is competitive in terms of observed accuracy while at the same time preserving the underlying positivity of the functions being interpolated.

math.NA↗

arXiv2022

Variational Inference for Nonlinear Inverse Problems via Neural Net Kernels: Comparison to Bayesian Neural Networks, Application to Topology Optimization

Inverse problems and, in particular, inferring unknown or latent parameters from data are ubiquitous in engineering simulations. A predominant viewpoint in identifying unknown parameters is Bayesian inference where both prior information about the parameters and the information from the observations via likelihood evaluations are incorporated into the inference process. In this paper, we adopt a similar viewpoint with a slightly different numerical procedure from standard inference approaches to provide insight about the localized behavior of unknown underlying parameters. We present a variational inference approach which mainly incorporates the observation data in a point-wise manner, i.e. we invert a limited number of observation data leveraging the gradient information of the forward map with respect to parameters, and find true individual samples of the latent parameters when the forward map is noise-free and one-to-one. For statistical calculations (as the ultimate goal in simulations), a large number of samples are generated from a trained neural network which serves as a transport map from the prior to posterior latent parameters. Our neural network machinery, developed as part of the inference framework and referred to as Neural Net Kernels (NNK), is based on hierarchical (deep) kernels which provide greater flexibility for training compared to standard neural networks. We showcase the effectiveness of our inference procedure in identifying bimodal and irregular distributions compared to a number of approaches including Markov Chain Monte Carlo sampling approaches and a Bayesian neural network approach.

stat.CO↗

arXiv2022

A bandit-learning approach to multifidelity approximation

Multifidelity approximation is an important technique in scientific computation and simulation. In this paper, we introduce a bandit-learning approach for leveraging data of varying fidelities to achieve precise estimates of the parameters of interest. Under a linear model assumption, we formulate a multifidelity approximation as a modified stochastic bandit, and analyze the loss for a class of policies that uniformly explore each model before exploiting. Utilizing the estimated conditional mean-squared error, we propose a consistent algorithm, adaptive Explore-Then-Commit (AETC), and establish a corresponding trajectory-wise optimality result. These results are then extended to the case of vector-valued responses, where we demonstrate that the algorithm is efficient without the need to worry about estimating high-dimensional parameters. The main advantage of our approach is that we require neither hierarchical model structure nor \textit{a priori} knowledge of statistical information (e.g., correlations) about or between models. Instead, the AETC algorithm requires only knowledge of which model is a trusted high-fidelity model, along with (relative) computational cost estimates of querying each model. Numerical experiments are provided at the end to support our theoretical findings.

math.NA↗

arXiv2022

Machine Learning in Heterogeneous Porous Materials

The "Workshop on Machine learning in heterogeneous porous materials" brought together international scientific communities of applied mathematics, porous media, and material sciences with experts in the areas of heterogeneous materials, machine learning (ML) and applied mathematics to identify how ML can advance materials research. Within the scope of ML and materials research, the goal of the workshop was to discuss the state-of-the-art in each community, promote crosstalk and accelerate multi-disciplinary collaborative research, and identify challenges and opportunities. As the end result, four topic areas were identified: ML in predicting materials properties, and discovery and design of novel materials, ML in porous and fractured media and time-dependent phenomena, Multi-scale modeling in heterogeneous porous materials via ML, and Discovery of materials constitutive laws and new governing equations. This workshop was part of the AmeriMech Symposium series sponsored by the National Academies of Sciences, Engineering and Medicine and the U.S. National Committee on Theoretical and Applied Mechanics.

cs.LG↗

arXiv2021

GP-HMAT: Scalable, ${O}(n\log(n))$ Gaussian Process Regression with Hierarchical Low-Rank Matrices

A Gaussian process (GP) is a powerful and widely used regression technique. The main building block of a GP regression is the covariance kernel, which characterizes the relationship between pairs in the random field. The optimization to find the optimal kernel, however, requires several large-scale and often unstructured matrix inversions. We tackle this challenge by introducing a hierarchical matrix approach, named HMAT, which effectively decomposes the matrix structure, in a recursive manner, into significantly smaller matrices where a direct approach could be used for inversion. Our matrix partitioning uses a particular aggregation strategy for data points, which promotes the low-rank structure of off-diagonal blocks in the hierarchical kernel matrix. We employ a randomized linear algebra method for matrix reduction on the low-rank off-diagonal blocks without factorizing a large matrix. We provide analytical error and cost estimates for the inversion of the matrix, investigate them empirically with numerical computations, and demonstrate the application of our approach on three numerical examples involving GP regression for engineering problems and a large-scale real dataset. We provide the computer implementation of GP-HMAT, HMAT adapted for GP likelihood and derivative computations, and the implementation of the last numerical example on a real dataset. We demonstrate superior scalability of the HMAT approach compared to built-in $\backslash$ operator in MATLAB for large-scale linear solves $\bf{A}\bf{x} = \bf{y}$ via a repeatable and verifiable empirical study. An extension to hierarchical semiseparable (HSS) matrices is discussed as future research.

math.NA↗

arXiv2021

Fast Barycentric-Based Evaluation Over Spectral/hp Elements

As the use of spectral/$hp$ element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendousattention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/$hp$ element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/$hp$ element library Nektar++.

math.NA↗

arXiv2021

Characterizing possible failure modes in physics-informed neural networks

Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN's loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training.

cs.LG↗

arXiv2021

Deep Multi-Fidelity Active Learning of High-dimensional Outputs

Many applications, such as in physical simulation and engineering design, demand we estimate functions with high-dimensional outputs. The training examples can be collected with different fidelities to allow a cost/accuracy trade-off. In this paper, we consider the active learning task that identifies both the fidelity and input to query new training examples so as to achieve the best benefit-cost ratio. To this end, we propose DMFAL, a Deep Multi-Fidelity Active Learning approach. We first develop a deep neural network-based multi-fidelity model for learning with high-dimensional outputs, which can flexibly, efficiently capture all kinds of complex relationships across the outputs and fidelities to improve prediction. We then propose a mutual information-based acquisition function that extends the predictive entropy principle. To overcome the computational challenges caused by large output dimensions, we use multi-variate Delta's method and moment-matching to estimate the output posterior, and Weinstein-Aronszajn identity to calculate and optimize the acquisition function. The computation is tractable, reliable and efficient. We show the advantage of our method in several applications of computational physics and engineering design.

cs.LG↗

arXiv2021

Batch Multi-Fidelity Bayesian Optimization with Deep Auto-Regressive Networks

Bayesian optimization (BO) is a powerful approach for optimizing black-box, expensive-to-evaluate functions. To enable a flexible trade-off between the cost and accuracy, many applications allow the function to be evaluated at different fidelities. In order to reduce the optimization cost while maximizing the benefit-cost ratio, in this paper, we propose Batch Multi-fidelity Bayesian Optimization with Deep Auto-Regressive Networks (BMBO-DARN). We use a set of Bayesian neural networks to construct a fully auto-regressive model, which is expressive enough to capture strong yet complex relationships across all the fidelities, so as to improve the surrogate learning and optimization performance. Furthermore, to enhance the quality and diversity of queries, we develop a simple yet efficient batch querying method, without any combinatorial search over the fidelities. We propose a batch acquisition function based on Max-value Entropy Search (MES) principle, which penalizes highly correlated queries and encourages diversity. We use posterior samples and moment matching to fulfill efficient computation of the acquisition function and conduct alternating optimization over every fidelity-input pair, which guarantees an improvement at each step. We demonstrate the advantage of our approach on four real-world hyperparameter optimization applications.

cs.LG↗

arXiv2021

Particle Merging-and-Splitting

Robustly handling collisions between individual particles in a large particle-based simulation has been a challenging problem. We introduce particle merging-and-splitting, a simple scheme for robustly handling collisions between particles that prevents inter-penetrations of separate objects without introducing numerical instabilities. This scheme merges colliding particles at the beginning of the time-step and then splits them at the end of the time-step. Thus, collisions last for the duration of a time-step, allowing neighboring particles of the colliding particles to influence each other. We show that our merging-and-splitting method is effective in robustly handling collisions and avoiding penetrations in particle-based simulations. We also show how our merging-and-splitting approach can be used for coupling different simulation systems using different and otherwise incompatible integrators. We present simulation tests involving complex solid-fluid interactions, including solid fractures generated by fluid interactions.

cs.GR↗