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Donsub Rim

Publications and source records attributed to Donsub Rim.

At least 19 recordsLinked to original sources
arXiv2026

Adversarial Training Without Input Gradients via Low-Rank Householder Expansions

This work concerns adversarial training against the small-norm adversarial examples that arise from the inherent input instability of a trained deep neural network. Examples in this class are small as measured in the relative $\ell^2$-norm, and therefore lie in the neighborhood of the input on which the model acts approximately linearly, the regime in which the perturbation remains imperceptible. We first show that such examples can be computed directly from the trained network parameters, without input gradient iterations, by means of a linearization called the low-rank Householder expansion (LRHE). The expansion describes the composed affine map rather than any individual layer, and the directions it identifies are read from the activation pattern already available in the forward pass. We then propose a simple adversarial training scheme built on this construction. No differentiation with respect to the input is performed at any point: training requires only additional forward evaluations, with weight parameters updated by the standard backward pass, and the inner maximization of the usual min-max formulation is eliminated entirely. That such a regularizer exists is our main finding: the methods that dispense with the inner search all obtain their local geometry by differentiating with respect to the input, and we show this is not necessary. The regularizer costs the equivalent of $2.8$ PGD steps per epoch, an $8.7\times$ reduction relative to 40-step adversarial training on MNIST and below the cost of 3-step training. The resulting models match three-step PGD adversarial training for relative $\ell^2$ budgets $\varepsilon \le 0.02$ and 40-step training for $\varepsilon \le 0.012$, falling away beyond, consistent with the locality of the expansion.

cs.LG↗
arXiv2026

Pulse-response analysis of a simple reaction-advection-diffusion equation

We analyze a reaction-advection-diffusion (RAD) equation arising in pulse-response studies of transport and reaction in a narrow reactor tube. A short pulse of gas is injected near one end of the reactor, while unreacted gas and reaction products are detected at the outlet. Particular attention is given to the effect of a constant axial advection velocity as a minimal extension of the standard diffusion model. For a single pulse, we obtain analytical expressions for the exit-flow response and for experimentally accessible characteristics, including its moments and peak properties, as functions of the P\'eclet number and the reaction rate. The corresponding diffusion-advection model without reaction defines a standard transport curve that may be used as a baseline for identifying chemical activity. For a first-order irreversible reaction, the reactive and nonreactive exit-flow curves satisfy a simple exponential factorization, allowing the reaction rate to be extracted directly from their ratio. We also extend the single-pulse analysis to a uniform periodic train of pulses. Using Poisson summation, we derive explicit expressions for the Fourier coefficients of the asymptotic periodic exit flow and relate them to the Laplace transform and moments of the single-pulse response. Finally, a stochastic interpretation in terms of reflected diffusion, first-passage times, and exponential killing provides probabilistic meaning for several of the analytical results and unifies the single-pulse, periodic-response, and reaction aspects of the model.

math.AP↗
arXiv2025

Learning Low Rank Neural Representations of Hyperbolic Wave Dynamics from Data

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.

cs.LG↗
arXiv2024

An explicit spectral decomposition of the ADRT

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.

math.NA↗
arXiv2024

FastLRNR and Sparse Physics Informed Backpropagation

We introduce Sparse Physics Informed Backpropagation (SPInProp), a new class of methods for accelerating backpropagation for a specialized neural network architecture called Low Rank Neural Representation (LRNR). The approach exploits the low rank structure within LRNR and constructs a reduced neural network approximation that is much smaller in size. We call the smaller network FastLRNR. We show that backpropagation of FastLRNR can be substituted for that of LRNR, enabling a significant reduction in complexity. We apply SPInProp to a physics informed neural networks framework and demonstrate how the solution of parametrized partial differential equations is accelerated.

cs.LG↗
arXiv2024

On the performance of sequential Bayesian update for database of diverse tsunami scenarios

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.

cs.CE↗
arXiv2024

A Low Rank Neural Representation of Entropy Solutions

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.

math.NA↗
arXiv2023

Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks

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.

cs.LG↗
arXiv2022

Mining Causality from Continuous-time Dynamics Models: An Application to Tsunami Forecasting

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.

cs.LG↗
arXiv2020

Manifold Approximations via Transported Subspaces: Model reduction for transport-dominated problems

This work presents a method for constructing online-efficient reduced models of large-scale systems governed by parametrized nonlinear scalar conservation laws. The solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures, which means that traditional model reduction methods based on linear approximations are inefficient when applied to these problems. In contrast, the approach introduced in this work derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. A time-stepping scheme evolves the nonlinear reduced models by transporting local approximation spaces along the characteristic curves of the governing equations. The proposed computational procedure allows an offline/online decomposition and is online-efficient in the sense that the complexity of accurately time-stepping the nonlinear reduced model is independent of that of the full model. Numerical experiments with transport through heterogeneous media and the Burgers' equation show orders of magnitude speedups of the proposed nonlinear reduced models based on transported subspaces compared to traditional linear reduced models and full models.

math.NA↗
arXiv2020

A range characterization of the single-quadrant ADRT

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.

math.NA↗
arXiv2020

Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems

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.

math.NA↗
arXiv2018

Model reduction of a parametrized scalar hyperbolic conservation law using displacement interpolation

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.

math.NA↗
arXiv2017

Displacement interpolation using monotone rearrangement

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].

math.NA↗
arXiv2017

Imaging of isotropic and anisotropic conductivities from power densities in three dimensions

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.

math.NA↗
arXiv2017

Dimensional splitting of hyperbolic partial differential equations using the Radon transform

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.

math.NA↗
arXiv2017

Transport reversal for model reduction of hyperbolic partial differential equations

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.

math.NA↗