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This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation's solution by its truncated Fourier expansion in the time domain and using a polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.

This paper addresses the benefits of pooling data for shared learning in maintenance operations. We consider a set of systems subject to Poisson degradation that are coupled through an a-priori unknown rate. Decision problems involving these systems are high-dimensional Markov decision processes (MDPs). We present a decomposition result that reduces such an MDP to two-dimensional MDPs, enabling structural analyses and computations. We leverage this decomposition to demonstrate that pooling data can lead to significant cost reductions compared to not pooling.

Accurately estimating parameters in complex nonlinear systems is crucial across scientific and engineering fields. We present a novel approach for parameter estimation using a neural network with the Huber loss function. This method taps into deep learning's abilities to uncover parameters governing intricate behaviors in nonlinear equations. We validate our approach using synthetic data and predefined functions that model system dynamics. By training the neural network with noisy time series data, it fine-tunes the Huber loss function to converge to accurate parameters. We apply our method to damped oscillators, Van der Pol oscillators, Lotka-Volterra systems, and Lorenz systems under multiplicative noise. The trained neural network accurately estimates parameters, evident from closely matching latent dynamics. Comparing true and estimated trajectories visually reinforces our method's precision and robustness. Our study underscores the Huber loss-guided neural network as a versatile tool for parameter estimation, effectively uncovering complex relationships in nonlinear systems. The method navigates noise and uncertainty adeptly, showcasing its adaptability to real-world challenges.

Replication studies are increasingly conducted to assess the credibility of scientific findings. Most of these replication attempts target studies with a superiority design, and there is a lack of methodology regarding the analysis of replication studies with alternative types of designs, such as equivalence. In order to fill this gap, we propose two approaches, the two-trials rule and the sceptical TOST procedure, adapted from methods used in superiority settings. Both methods have the same overall Type-I error rate, but the sceptical TOST procedure allows replication success even for non-significant original or replication studies. This leads to a larger project power and other differences in relevant operating characteristics. Both methods can be used for sample size calculation of the replication study, based on the results from the original one. The two methods are applied to data from the Reproducibility Project: Cancer Biology.

The real-time dynamic environment perception has become vital for autonomous robots in crowded spaces. Although the popular voxel-based mapping methods can efficiently represent 3D obstacles with arbitrarily complex shapes, they can hardly distinguish between static and dynamic obstacles, leading to the limited performance of obstacle avoidance. While plenty of sophisticated learning-based dynamic obstacle detection algorithms exist in autonomous driving, the quadcopter's limited computation resources cannot achieve real-time performance using those approaches. To address these issues, we propose a real-time dynamic obstacle tracking and mapping system for quadcopter obstacle avoidance using an RGB-D camera. The proposed system first utilizes a depth image with an occupancy voxel map to generate potential dynamic obstacle regions as proposals. With the obstacle region proposals, the Kalman filter and our continuity filter are applied to track each dynamic obstacle. Finally, the environment-aware trajectory prediction method is proposed based on the Markov chain using the states of tracked dynamic obstacles. We implemented the proposed system with our custom quadcopter and navigation planner. The simulation and physical experiments show that our methods can successfully track and represent obstacles in dynamic environments in real-time and safely avoid obstacles.

The burden of liver tumors is important, ranking as the fourth leading cause of cancer mortality. In case of hepatocellular carcinoma (HCC), the delineation of liver and tumor on contrast-enhanced magnetic resonance imaging (CE-MRI) is performed to guide the treatment strategy. As this task is time-consuming, needs high expertise and could be subject to inter-observer variability there is a strong need for automatic tools. However, challenges arise from the lack of available training data, as well as the high variability in terms of image resolution and MRI sequence. In this work we propose to compare two different pipelines based on anisotropic models to obtain the segmentation of the liver and tumors. The first pipeline corresponds to a baseline multi-class model that performs the simultaneous segmentation of the liver and tumor classes. In the second approach, we train two distinct binary models, one segmenting the liver only and the other the tumors. Our results show that both pipelines exhibit different strengths and weaknesses. Moreover we propose an uncertainty quantification strategy allowing the identification of potential false positive tumor lesions. Both solutions were submitted to the MICCAI 2023 Atlas challenge regarding liver and tumor segmentation.

Sequential recommendation involves automatically recommending the next item to users based on their historical item sequence. While most prior research employs RNN or transformer methods to glean information from the item sequence-generating probabilities for each user-item pair and recommending the top items, these approaches often overlook the challenge posed by spurious relationships. This paper specifically addresses these spurious relations. We introduce a novel sequential recommendation framework named Irl4Rec. This framework harnesses invariant learning and employs a new objective that factors in the relationship between spurious variables and adjustment variables during model training. This approach aids in identifying spurious relations. Comparative analyses reveal that our framework outperforms three typical methods, underscoring the effectiveness of our model. Moreover, an ablation study further demonstrates the critical role our model plays in detecting spurious relations.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.

The Evidential regression network (ENet) estimates a continuous target and its predictive uncertainty without costly Bayesian model averaging. However, it is possible that the target is inaccurately predicted due to the gradient shrinkage problem of the original loss function of the ENet, the negative log marginal likelihood (NLL) loss. In this paper, the objective is to improve the prediction accuracy of the ENet while maintaining its efficient uncertainty estimation by resolving the gradient shrinkage problem. A multi-task learning (MTL) framework, referred to as MT-ENet, is proposed to accomplish this aim. In the MTL, we define the Lipschitz modified mean squared error (MSE) loss function as another loss and add it to the existing NLL loss. The Lipschitz modified MSE loss is designed to mitigate the gradient conflict with the NLL loss by dynamically adjusting its Lipschitz constant. By doing so, the Lipschitz MSE loss does not disturb the uncertainty estimation of the NLL loss. The MT-ENet enhances the predictive accuracy of the ENet without losing uncertainty estimation capability on the synthetic dataset and real-world benchmarks, including drug-target affinity (DTA) regression. Furthermore, the MT-ENet shows remarkable calibration and out-of-distribution detection capability on the DTA benchmarks.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.