Donsub Rim

Donsub Rim, Scott Moe, Randall J. LeVeque

Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem, we build on previous work in symmetry reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterative algorithm that decomposes the snapshot matrix into multiple shifting profiles, each with a corresponding speed. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Finally, we give a geometric interpretation of the algorithm.

Randall J. LeVeque, Knut Waagan, Frank I. González et al.

In order to perform probabilistic tsunami hazard assessment (PTHA) based on subduction zone earthquakes, it is necessary to start with a catalog of possible future events along with the annual probability of occurance, or a probability distribution of such events that can be easily sampled. For nearfield events, the distribution of slip on the fault can have a significant effect on the resulting tsunami. We present an approach to defining a probability distribution based on subdividing the fault geometry into many subfaults and prescribing a desired covariance matrix relating slip on one subfault to slip on any other subfault. The eigenvalues and eigenvectors of this matrix are then used to define a Karhunen-Loève expansion for random slip patterns. This is similar to a spectral representation of random slip based on Fourier series but conforms to a general fault geometry. We show that only a few terms in this series are needed to represent the features of the slip distribution that are most important in tsunami generation, first with a simple one-dimensional example where slip varies only in the down-dip direction and then on a portion of the Cascadia Subduction Zone.

Donsub Rim, Kyle T. Mandli

We propose a model reduction technique for parametrized partial differential equations arising from scalar hyperbolic conservation laws. The key idea of the technique is to construct basis functions that are local in parameter and time space via displacement interpolation. The construction is motivated by the observation that the derivative of solutions to hyperbolic conservation laws satisfy a contractive property with respect to the Wasserstein metric [Bolley et al. J. Hyperbolic Differ. Equ. 02 (2005), pp. 91-107]. We will discuss the approximation properties of the displacement interpolation, and show that it can naturally complement linear interpolation. Numerical experiments illustrate that we can successfully achieve the model reduction of a parametrized Burgers' equation, and that the reduced order model is suitable for performing typical tasks in uncertainty quantification.

Woojin Cho, Kookjin Lee, Donsub Rim et al.

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design parameters) and solvers are required to perform rapid execution. In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs), emerging deep-learning-based solvers, to be considered as one such solver. Although PINNs have pioneered a proper integration of deep-learning and scientific computing, they require repetitive time-consuming training of neural networks, which is not suitable for many-query scenarios. To address this issue, we propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allows efficient approximation of solutions of PDEs for varying ranges of PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as "failure modes" of PINNs.

François Monard, Donsub Rim

We present numerical reconstructions of anisotropic conductivity tensors in three dimensions, from knowledge of a finite family of power density functionals. Such a problem arises in the coupled-physics imaging modality Ultrasound Modulated Electrical Impedance Tomography for instance. We improve on the algorithms previously derived in [Bal et al, Inverse Probl Imaging (2013), pp.353-375, Monard and Bal, Comm. PDE (2013), pp.1183-1207] for both isotropic and anisotropic cases, and we address the well-known issue of vanishing determinants in particular. The algorithm is implemented and we provide numerical results that illustrate the improvements.

Fan Wu, Sanghyun Hong, Donsub Rim et al.

Continuous-time dynamics models, such as neural ordinary differential equations, have enabled the modeling of underlying dynamics in time-series data and accurate forecasting. However, parameterization of dynamics using a neural network makes it difficult for humans to identify causal structures in the data. In consequence, this opaqueness hinders the use of these models in the domains where capturing causal relationships carries the same importance as accurate predictions, e.g., tsunami forecasting. In this paper, we address this challenge by proposing a mechanism for mining causal structures from continuous-time models. We train models to capture the causal structure by enforcing sparsity in the weights of the input layers of the dynamics models. We first verify the effectiveness of our method in the scenario where the exact causal-structures of time-series are known as a priori. We next apply our method to a real-world problem, namely tsunami forecasting, where the exact causal-structures are difficult to characterize. Experimental results show that the proposed method is effective in learning physically-consistent causal relationships while achieving high forecasting accuracy.

Donsub Rim

We introduce a dimensional splitting method based on the intertwining property of the Radon transform, with a particular focus on its applications related to hyperbolic partial differential equations (PDEs). This dimensional splitting has remarkable properties that makes it useful in a variety of contexts, including multi-dimensional extension of large time-step (LTS) methods, absorbing boundary conditions, displacement interpolation, and multi-dimensional generalization of transport reversal.

Reika Nomura, Louise A. Hirao Vermare, Saneiki Fujita et al.

Although the sequential tsunami scenario detection framework was validated in our previous work, several tasks remain to be resolved from a practical point of view. This study aims to evaluate the performance of the previous tsunami scenario detection framework using a diverse database consisting of complex fault rupture patterns with heterogeneous slip distributions. Specifically, we compare the effectiveness of scenario superposition to that of the previous most likely scenario detection method. Additionally, how the length of the observation time window influences the accuracy of both methods is analyzed. We utilize an existing database comprising 1771 tsunami scenarios targeting the city Westport (WA, U.S.), which includes synthetic wave height records and inundation distributions as the result of fault rupture in the Cascadia subduction zone. The heterogeneous patterns of slips used in the database increase the diversity of the scenarios and thus make it a proper database for evaluating the performance of scenario superposition. To assess the performance, we consider various observation time windows shorter than 15 minutes and divide the database into five testing and learning sets. The evaluation accuracy of the maximum offshore wave, inundation depth, and its distribution is analyzed to examine the advantages of the scenario superposition method over the previous method. We introduce the dynamic time warping (DTW) method as an additional benchmark and compare its results to that of the Bayesian scenario detection method.

Woojin Cho, Kookjin Lee, Noseong Park et al.

We present a data-driven dimensionality reduction method that is well-suited for physics-based data representing hyperbolic wave propagation. The method utilizes a specialized neural network architecture called low rank neural representation (LRNR) inside a hypernetwork framework. The architecture is motivated by theoretical results that rigorously prove the existence of efficient representations for this wave class. We illustrate through archetypal examples that such an efficient low-dimensional representation of propagating waves can be learned directly from data through a combination of deep learning techniques. We observe that a low rank tensor representation arises naturally in the trained LRNRs, and that this reveals a new decomposition of wave propagation where each decomposed mode corresponds to interpretable physical features. Furthermore, we demonstrate that the LRNR architecture enables efficient inference via a compression scheme, which is a potentially important feature when deploying LRNRs in demanding performance regimes.

Donsub Rim, Gerrit Welper

We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.

Weilin Li, Kui Ren, Donsub Rim

This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization follows from a set of linear constraints on the codomain. We show that for data satisfying these constraints, the exact and fast inversion formula [Rim, Appl. Math. Lett. 102 106159, 2020] yields a square image in a stable manner. The range characterization is obtained by first showing that the ADRT is a bijection between images supported on infinite half-strips, then identifying the linear subspaces that stay finitely supported under the inversion formula.

Donsub Rim, Luca Venturi, Joan Bruna et al.

Classical reduced models are low-rank approximations using a fixed basis designed to achieve dimensionality reduction of large-scale systems. In this work, we introduce reduced deep networks, a generalization of classical reduced models formulated as deep neural networks. We prove depth separation results showing that reduced deep networks approximate solutions of parametrized hyperbolic partial differential equations with approximation error $ε$ with $\mathcal{O}(|\log(ε)|)$ degrees of freedom, even in the nonlinear setting where solutions exhibit shock waves. We also show that classical reduced models achieve exponentially worse approximation rates by establishing lower bounds on the relevant Kolmogorov $N$-widths.

Donsub Rim

We give an exact inversion formula for the approximate discrete Radon transform introduced in [Brady, SIAM J. Comput., 27(1), 107--119] that is of cost $O(N \log N)$ for a square 2D image with $N$ pixels and requires only partial data.

Donsub Rim, Kyle T. Mandli

When approximating a function that depends on a parameter, one encounters many practical examples where linear interpolation or linear approximation with respect to the parameters prove ineffective. This is particularly true for responses from hyperbolic partial differential equations (PDEs) where linear, low-dimensional bases are difficult to construct. We propose the use of displacement interpolation where the interpolation is done on the optimal transport map between the functions at nearby parameters, to achieve an effective dimensionality reduction of hyperbolic phenomena. We further propose a multi-dimensional extension by using the intertwining property of the Radon transform. This extension is a generalization of the classical translational representation of Lax-Philips [Lax and Philips, Bull. Amer. Math. Soc. 70 (1964), pp.130--142].