A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658x speed-up with 1 to 5% relative errors.
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The generative autoencoders, such as the variational autoencoders or the adversarial autoencoders, have achieved great success in lots of real-world applications, including image generation, and signal communication. However, little concern has been devoted to their robustness during practical deployment. Due to the probabilistic latent structure, variational autoencoders (VAEs) may confront problems such as a mismatch between the posterior distribution of the latent and real data manifold, or discontinuity in the posterior distribution of the latent. This leaves a back door for malicious attackers to collapse VAEs from the latent space, especially in scenarios where the encoder and decoder are used separately, such as communication and compressed sensing. In this work, we provide the first study on the adversarial robustness of generative autoencoders in the latent space. Specifically, we empirically demonstrate the latent vulnerability of popular generative autoencoders through attacks in the latent space. We also evaluate the difference between variational autoencoders and their deterministic variants and observe that the latter performs better in latent robustness. Meanwhile, we identify a potential trade-off between the adversarial robustness and the degree of the disentanglement of the latent codes. Additionally, we also verify the feasibility of improvement for the latent robustness of VAEs through adversarial training. In summary, we suggest concerning the adversarial latent robustness of the generative autoencoders, analyze several robustness-relative issues, and give some insights into a series of key challenges.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.
It is challenging to perform identification on soft robots due to their underactuated, high dimensional dynamics. In this work, we present a data-driven modeling framework, based on geometric mechanics (also known as gauge theory), that can be applied to systems with low-bandwidth actuation of the shape space. By exploiting temporal asymmetries in actuator dynamics, our approach enables the design of robots that can be driven by a single control input. We present a method for constructing a series connected model comprising actuator and locomotor dynamics based on data points from stochastically perturbed, repeated behaviors around the observed limit cycle. We demonstrate our methods on a real-world example of a soft crawler made by stimuli-responsive hydrogels that locomotes on merely one cycling control signal by utilizing its geometric and temporal asymmetry. For systems with first-order, low-pass actuator dynamics, such as swelling-driven actuators used in hydrogel crawlers, we show that first order Taylor approximations can well capture the dynamics of the system shape as well as its movements. Finally, we propose an approach of numerically optimizing control signals by iteratively refining models and optimizing the input waveform.
Physics-informed neural networks (PINNs) have recently emerged as promising data-driven PDE solvers showing encouraging results on various PDEs. However, there is a fundamental limitation of training PINNs to solve multi-dimensional PDEs and approximate highly complex solution functions. The number of training points (collocation points) required on these challenging PDEs grows substantially, but it is severely limited due to the expensive computational costs and heavy memory overhead. To overcome this issue, we propose a network architecture and training algorithm for PINNs. The proposed method, separable PINN (SPINN), operates on a per-axis basis to significantly reduce the number of network propagations in multi-dimensional PDEs unlike point-wise processing in conventional PINNs. We also propose using forward-mode automatic differentiation to reduce the computational cost of computing PDE residuals, enabling a large number of collocation points (>10^7) on a single commodity GPU. The experimental results show drastically reduced computational costs (62x in wall-clock time, 1,394x in FLOPs given the same number of collocation points) in multi-dimensional PDEs while achieving better accuracy. Furthermore, we present that SPINN can solve a chaotic (2+1)-d Navier-Stokes equation significantly faster than the best-performing prior method (9 minutes vs 10 hours in a single GPU), maintaining accuracy. Finally, we showcase that SPINN can accurately obtain the solution of a highly nonlinear and multi-dimensional PDE, a (3+1)-d Navier-Stokes equation. For visualized results and code, please see //jwcho5576.github.io/spinn.github.io/.
In classification problems, the datasets are usually imbalanced, noisy or complex. Most sampling algorithms only make some improvements to the linear sampling mechanism of the synthetic minority oversampling technique (SMOTE). Nevertheless, linear oversampling has several unavoidable drawbacks. Linear oversampling is susceptible to overfitting, and the synthetic samples lack diversity and rarely account for the original distribution characteristics. An informed nonlinear oversampling framework with the granular ball (INGB) as a new direction of oversampling is proposed in this paper. It uses granular balls to simulate the spatial distribution characteristics of datasets, and informed entropy is utilized to further optimize the granular-ball space. Then, nonlinear oversampling is performed by following high-dimensional sparsity and the isotropic Gaussian distribution. Furthermore, INGB has good compatibility. Not only can it be combined with most SMOTE-based sampling algorithms to improve their performance, but it can also be easily extended to noisy imbalanced multi-classification problems. The mathematical model and theoretical proof of INGB are given in this work. Extensive experiments demonstrate that INGB outperforms the traditional linear sampling frameworks and algorithms in oversampling on complex datasets.
This paper focuses on the quasi-optimality of an adaptive nonconforming FEM for a distributed optimal control problem governed by the Stokes equations. The nonconforming lowest order Crouzeix-Raviart element and piecewise constant spaces are used to discretize the velocity and pressure variables, respectively. The control variable is discretized using a variational approach. We present error equivalence results at both continuous and discrete levels, leading to a priori and a posteriori error estimates. The quasi-optimal convergence rates of the adaptive algorithm are established based on a general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality.
Multi-flagellated bacteria utilize the hydrodynamic interaction between their filamentary tails, known as flagella, to swim and change their swimming direction in low Reynolds number flow. This interaction, referred to as bundling and tumbling, is often overlooked in simplified hydrodynamic models such as Resistive Force Theories (RFT). However, for the development of efficient and steerable robots inspired by bacteria, it becomes crucial to exploit this interaction. In this paper, we present the construction of a macroscopic bio-inspired robot featuring two rigid flagella arranged as right-handed helices, along with a cylindrical head. By rotating the flagella in opposite directions, the robot's body can reorient itself through repeatable and controllable tumbling. To accurately model this bi-flagellated mechanism in low Reynolds flow, we employ a coupling of rigid body dynamics and the method of Regularized Stokeslet Segments (RSS). Unlike RFT, RSS takes into account the hydrodynamic interaction between distant filamentary structures. Furthermore, we delve into the exploration of the parameter space to optimize the propulsion and torque of the system. To achieve the desired reorientation of the robot, we propose a tumble control scheme that involves modulating the rotation direction and speed of the two flagella. By implementing this scheme, the robot can effectively reorient itself to attain the desired attitude. Notably, the overall scheme boasts a simplified design and control as it only requires two control inputs. With our macroscopic framework serving as a foundation, we envision the eventual miniaturization of this technology to construct mobile and controllable micro-scale bacterial robots.
We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative preconditioners significantly improve the convergence of the standard L-BFGS optimizer, while providing more accurate solutions of the underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism.
Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.
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