Michael Penwarden, Ameya D. Jagtap, Shandian Zhe et al.
Physics-informed neural networks (PINNs) as a means of solving partial differential equations (PDE) have garnered much attention in the Computational Science and Engineering (CS&E) world. However, a recent topic of interest is exploring various training (i.e., optimization) challenges - in particular, arriving at poor local minima in the optimization landscape results in a PINN approximation giving an inferior, and sometimes trivial, solution when solving forward time-dependent PDEs with no data. This problem is also found in, and in some sense more difficult, with domain decomposition strategies such as temporal decomposition using XPINNs. We furnish examples and explanations for different training challenges, their cause, and how they relate to information propagation and temporal decomposition. We then propose a new stacked-decomposition method that bridges the gap between time-marching PINNs and XPINNs. We also introduce significant computational speed-ups by using transfer learning concepts to initialize subnetworks in the domain and loss tolerance-based propagation for the subdomains. Finally, we formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality literature, which our framework can still describe, and provides a significant computational speed-up via reduced-cost collocation point segmentation. The proposed methods form our unified framework, which overcomes training challenges in PINNs and XPINNs for time-dependent PDEs by respecting the causality in multiple forms and improving scalability by limiting the computation required per optimization iteration. Finally, we provide numerical results for these methods on baseline PDE problems for which unmodified PINNs and XPINNs struggle to train.
Practical tensor data is often along with time information. Most existing temporal decomposition approaches estimate a set of fixed factors for the objects in each tensor mode, and hence cannot capture the temporal evolution of the objects' representation. More important, we lack an effective approach to capture such evolution from streaming data, which is common in real-world applications. To address these issues, we propose Streaming Factor Trajectory Learning for temporal tensor decomposition. We use Gaussian processes (GPs) to model the trajectory of factors so as to flexibly estimate their temporal evolution. To address the computational challenges in handling streaming data, we convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE). We develop an efficient online filtering algorithm to estimate a decoupled running posterior of the involved factor states upon receiving new data. The decoupled estimation enables us to conduct standard Rauch-Tung-Striebel smoothing to compute the full posterior of all the trajectories in parallel, without the need for revisiting any previous data. We have shown the advantage of SFTL in both synthetic tasks and real-world applications. The code is available at {https://github.com/xuangu-fang/Streaming-Factor-Trajectory-Learning}.
We present a new algorithm for the automatic one-shot generation of scattered node sets on irregular 2D and 3D domains using Poisson disk sampling coupled to novel parameter-free, high-order parametric Spherical Radial Basis Function (SBF)-based geometric modeling of irregular domain boundaries. Our algorithm also automatically modifies the scattered node sets locally for time-varying embedded boundaries in the domain interior. We derive complexity estimates for our node generator in 2D and 3D that establish its scalability, and verify these estimates with timing experiments. We explore the influence of Poisson disk sampling parameters on both quasi-uniformity in the node sets and errors in an RBF-FD discretization of the heat equation. In all cases, our framework requires only a small number of "seed" nodes on domain boundaries. The entire framework exhibits O(N) complexity in both 2D and 3D.
Julia Docampo Sánchez, Jennifer K. Ryan, Mahsa Mirzargar et al.
Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. These advantages can be exploited during flow visualization, for example by applying the SIAC filter to the DG data before streamline computations [Steffan {\it et al.}, IEEE-TVCG 14(3): 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [Walfisch {\it et al.}, JOMP 38(2);164-184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the SIAC line filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.
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.
Vidhi Zala, Varun Shankar, Shankar P. Sastry et al.
We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.
In this paper, we present a machine learning method for the discovery of analytic solutions to differential equations. The method utilizes an inherently interpretable algorithm, genetic programming based symbolic regression. Unlike conventional accuracy measures in machine learning we demonstrate the ability to recover true analytic solutions, as opposed to a numerical approximation. The method is verified by assessing its ability to recover known analytic solutions for two separate differential equations. The developed method is compared to a conventional, purely data-driven genetic programming based symbolic regression algorithm. The reliability of successful evolution of the true solution, or an algebraic equivalent, is demonstrated.
Da Long, Wei W. Xing, Aditi S. Krishnapriyan et al.
Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity and noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior -- an ideal Bayesian sparse distribution -- for effective operator selection and uncertainty quantification. We develop an expectation-propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra to enable efficient computation and optimization. We show the advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.
Zachary Bastiani, Robert M. Kirby, Jacob Hochhalter et al.
Diffusion has emerged as a powerful framework for generative modeling, achieving remarkable success in applications such as image and audio synthesis. Enlightened by this progress, we propose a novel diffusion-based approach for symbolic regression. We construct a random mask-based diffusion and denoising process to generate diverse and high-quality equations. We integrate this generative processes with a token-wise Group Relative Policy Optimization (GRPO) method to conduct efficient reinforcement learning on the given measurement dataset. In addition, we introduce a long short-term risk-seeking policy to expand the pool of top-performing candidates, further enhancing performance. Extensive experiments and ablation studies have demonstrated the effectiveness of our approach.
Zachary Bastiani, Robert M. Kirby, Jacob Hochhalter et al.
We propose a novel deep symbolic regression approach to enhance the robustness and interpretability of data-driven mathematical expression discovery. Our work is aligned with the popular DSR framework which focuses on learning a data-specific expression generator, without relying on pretrained models or additional search or planning procedures. Despite the success of existing DSR methods, they are built on recurrent neural networks, solely guided by data fitness, and potentially meet tail barriers that can zero out the policy gradient, causing inefficient model updates. To overcome these limitations, we design a decoder-only architecture that performs attention in the frequency domain and introduce a dual-indexed position encoding to conduct layer-wise generation. Second, we propose a Bayesian information criterion (BIC)-based reward function that can automatically adjust the trade-off between expression complexity and data fitness, without the need for explicit manual tuning. Third, we develop a ranking-based weighted policy update method that eliminates the tail barriers and enhances training effectiveness. Extensive benchmarks and systematic experiments demonstrate the advantages of our approach.
Model Agnostic Meta Learning (MAML) is widely used to find a good initialization for a family of tasks. Despite its success, a critical challenge in MAML is to calculate the gradient w.r.t. the initialization of a long training trajectory for the sampled tasks, because the computation graph can rapidly explode and the computational cost is very expensive. To address this problem, we propose Adjoint MAML (A-MAML). We view gradient descent in the inner optimization as the evolution of an Ordinary Differential Equation (ODE). To efficiently compute the gradient of the validation loss w.r.t. the initialization, we use the adjoint method to construct a companion, backward ODE. To obtain the gradient w.r.t. the initialization, we only need to run the standard ODE solver twice -- one is forward in time that evolves a long trajectory of gradient flow for the sampled task; the other is backward and solves the adjoint ODE. We need not create or expand any intermediate computational graphs, adopt aggressive approximations, or impose proximal regularizers in the training loss. Our approach is cheap, accurate, and adaptable to different trajectory lengths. We demonstrate the advantage of our approach in both synthetic and real-world meta-learning tasks.
Robert Kirby, Kolby Nottingham, Rajarshi Roy et al.
Recent advances in GPU accelerated global and detail placement have reduced the time to solution by an order of magnitude. This advancement allows us to leverage data driven optimization (such as Reinforcement Learning) in an effort to improve the final quality of placement results. In this work we augment state-of-the-art, force-based global placement solvers with a reinforcement learning agent trained to improve the final detail placed Half Perimeter Wire Length (HPWL). We propose novel control schemes with either global or localized control of the placement process. We then train reinforcement learning agents to use these controls to guide placement to improved solutions. In both cases, the augmented optimizer finds improved placement solutions. Our trained agents achieve an average 1% improvement in final detail place HPWL across a range of academic benchmarks and more than 1% in global place HPWL on real industry designs.
Michael Penwarden, Shandian Zhe, Akil Narayan et al.
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.
The key task of physical simulation is to solve partial differential equations (PDEs) on discretized domains, which is known to be costly. In particular, high-fidelity solutions are much more expensive than low-fidelity ones. To reduce the cost, we consider novel Gaussian process (GP) models that leverage simulation examples of different fidelities to predict high-dimensional PDE solution outputs. Existing GP methods are either not scalable to high-dimensional outputs or lack effective strategies to integrate multi-fidelity examples. To address these issues, we propose Multi-Fidelity High-Order Gaussian Process (MFHoGP) that can capture complex correlations both between the outputs and between the fidelities to enhance solution estimation, and scale to large numbers of outputs. Based on a novel nonlinear coregionalization model, MFHoGP propagates bases throughout fidelities to fuse information, and places a deep matrix GP prior over the basis weights to capture the (nonlinear) relationships across the fidelities. To improve inference efficiency and quality, we use bases decomposition to largely reduce the model parameters, and layer-wise matrix Gaussian posteriors to capture the posterior dependency and to simplify the computation. Our stochastic variational learning algorithm successfully handles millions of outputs without extra sparse approximations. We show the advantages of our method in several typical applications.
We present an approach for high-resolution video frame prediction by conditioning on both past frames and past optical flows. Previous approaches rely on resampling past frames, guided by a learned future optical flow, or on direct generation of pixels. Resampling based on flow is insufficient because it cannot deal with disocclusions. Generative models currently lead to blurry results. Recent approaches synthesis a pixel by convolving input patches with a predicted kernel. However, their memory requirement increases with kernel size. Here, we spatially-displaced convolution (SDC) module for video frame prediction. We learn a motion vector and a kernel for each pixel and synthesize a pixel by applying the kernel at a displaced location in the source image, defined by the predicted motion vector. Our approach inherits the merits of both vector-based and kernel-based approaches, while ameliorating their respective disadvantages. We train our model on 428K unlabelled 1080p video game frames. Our approach produces state-of-the-art results, achieving an SSIM score of 0.904 on high-definition YouTube-8M videos, 0.918 on Caltech Pedestrian videos. Our model handles large motion effectively and synthesizes crisp frames with consistent motion.
Recent work has shown how to train Convolutional Neural Networks (CNNs) rapidly on large image datasets, then transfer the knowledge gained from these models to a variety of tasks. Following [Radford 2017], in this work, we demonstrate similar scalability and transfer for Recurrent Neural Networks (RNNs) for Natural Language tasks. By utilizing mixed precision arithmetic and a 32k batch size distributed across 128 NVIDIA Tesla V100 GPUs, we are able to train a character-level 4096-dimension multiplicative LSTM (mLSTM) for unsupervised text reconstruction over 3 epochs of the 40 GB Amazon Reviews dataset in four hours. This runtime compares favorably with previous work taking one month to train the same size and configuration for one epoch over the same dataset. Converging large batch RNN models can be challenging. Recent work has suggested scaling the learning rate as a function of batch size, but we find that simply scaling the learning rate as a function of batch size leads either to significantly worse convergence or immediate divergence for this problem. We provide a learning rate schedule that allows our model to converge with a 32k batch size. Since our model converges over the Amazon Reviews dataset in hours, and our compute requirement of 128 Tesla V100 GPUs, while substantial, is commercially available, this work opens up large scale unsupervised NLP training to most commercial applications and deep learning researchers. A model can be trained over most public or private text datasets overnight.