Simulating rare events, such as the transformation of a reactant into a product in a chemical reaction typically requires enhanced sampling techniques that rely on heuristically chosen collective variables (CVs). We propose using differentiable simulations (DiffSim) for the discovery and enhanced sampling of chemical transformations without a need to resort to preselected CVs, using only a distance metric. Reaction path discovery and estimation of the biasing potential that enhances the sampling are merged into a single end-to-end problem that is solved by path-integral optimization. This is achieved by introducing multiple improvements over standard DiffSim such as partial backpropagation and graph mini-batching making DiffSim training stable and efficient. The potential of DiffSim is demonstrated in the successful discovery of transition paths for the Muller-Brown model potential as well as a benchmark chemical system - alanine dipeptide.
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A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure $p$ and a non linear reaction-diffusion equation for the conductivity tensor $\mathbb{C}$. The problem requires high resolution due to the presence of multiple scales, the stiffness in all its components and the non linearities. We propose a low order finite element discretization in space coupled with a semi-implicit time advancing scheme. The code is validated with several numerical tests performed with various choices for the parameters involved in the system. In absence of the exact solution, we apply Richardson extrapolation technique to estimate the order of the method.
Model-reference adaptive systems refer to a consortium of techniques that guide plants to track desired reference trajectories. Approaches based on theories like Lyapunov, sliding surfaces, and backstepping are typically employed to advise adaptive control strategies. The resulting solutions are often challenged by the complexity of the reference model and those of the derived control strategies. Additionally, the explicit dependence of the control strategies on the process dynamics and reference dynamical models may contribute in degrading their efficiency in the face of uncertain or unknown dynamics. A model-reference adaptive solution is developed here for autonomous systems where it solves the Hamilton-Jacobi-Bellman equation of an error-based structure. The proposed approach describes the process with an integral temporal difference equation and solves it using an integral reinforcement learning mechanism. This is done in real-time without knowing or employing the dynamics of either the process or reference model in the control strategies. A class of aircraft is adopted to validate the proposed technique.
We study learning in periodic Markov Decision Process (MDP), a special type of non-stationary MDP where both the state transition probabilities and reward functions vary periodically, under the average reward maximization setting. We formulate the problem as a stationary MDP by augmenting the state space with the period index, and propose a periodic upper confidence bound reinforcement learning-2 (PUCRL2) algorithm. We show that the regret of PUCRL2 varies linearly with the period $N$ and as $\mathcal{O}(\sqrt{Tlog T})$ with the horizon length $T$. Utilizing the information about the sparsity of transition matrix of augmented MDP, we propose another algorithm PUCRLB which enhances upon PUCRL2, both in terms of regret ($O(\sqrt{N})$ dependency on period) and empirical performance. Finally, we propose two other algorithms U-PUCRL2 and U-PUCRLB for extended uncertainty in the environment in which the period is unknown but a set of candidate periods are known. Numerical results demonstrate the efficacy of all the algorithms.
Deep neural networks (DNNs) have emerged as a dominant approach for developing traffic forecasting models. These models are typically trained to minimize error on averaged test cases and produce a single-point prediction, such as a scalar value for traffic speed or travel time. However, single-point predictions fail to account for prediction uncertainty that is critical for many transportation management scenarios, such as determining the best- or worst-case arrival time. We present QuanTraffic, a generic framework to enhance the capability of an arbitrary DNN model for uncertainty modeling. QuanTraffic requires little human involvement and does not change the base DNN architecture during deployment. Instead, it automatically learns a standard quantile function during the DNN model training to produce a prediction interval for the single-point prediction. The prediction interval defines a range where the true value of the traffic prediction is likely to fall. Furthermore, QuanTraffic develops an adaptive scheme that dynamically adjusts the prediction interval based on the location and prediction window of the test input. We evaluated QuanTraffic by applying it to five representative DNN models for traffic forecasting across seven public datasets. We then compared QuanTraffic against five uncertainty quantification methods. Compared to the baseline uncertainty modeling techniques, QuanTraffic with base DNN architectures delivers consistently better and more robust performance than the existing ones on the reported datasets.
Consider a hiring process with candidates coming from different universities. It is easy to order candidates who have the exact same background, yet it can be challenging to compare candidates otherwise. The latter case requires additional assessments, leading to a potentially high total cost for the hiring organization. Given an assigned budget, what is the optimal strategy to select the most qualified candidate? In the absence of additional information, we model the above problem by introducing a new variant of the secretary problem. Completely ordered candidates, belonging to distinct groups, are arriving in a sequential manner. The decision maker has access to the partial order of the candidates within their own group and can request access to the total order of observed candidates by paying some price. Given a bounded budget of comparisons, the goal of the decision-maker is to maximize the probability of selecting the best candidate. We consider a special case of two groups with stochastic i.i.d.\ group membership. We introduce and analyze a particular family of algorithms that we called Dynamic Double Threshold (DDT) family, deriving its asymptotic success probability which, given an optimal choice of parameter converges rapidly to the theoretical upper bound of $1/e$ as the comparison budget growth. We provide an optimal non-asymptotic memory-less algorithm for the above problem and give numerical evidence that it belongs to the DDT family when the number of candidates is high. We compare theoretically and numerically the optimal algorithm with a more naive approach that is directly inspired by the standard single-threshold secretary algorithm. Our analysis reveals several alluring properties of the optimal algorithm. It provides a step towards a fairer online selection process in the presence of unidentifiable biases.
We present DiffXPBD, a novel and efficient analytical formulation for the differentiable position-based simulation of compliant constrained dynamics (XPBD). Our proposed method allows computation of gradients of numerous parameters with respect to a goal function simultaneously leveraging a performant simulation model. The method is efficient, thus enabling differentiable simulations of high resolution geometries and degrees of freedom (DoFs). Collisions are naturally included in the framework. Our differentiable model allows a user to easily add additional optimization variables. Every control variable gradient requires the computation of only a few partial derivatives which can be computed using automatic differentiation code. We demonstrate the efficacy of the method with examples such as elastic material parameter estimation, initial value optimization, optimizing for underlying body shape and pose by only observing the clothing, and optimizing a time-varying external force sequence to match sparse keyframe shapes at specific times. Our approach demonstrates excellent efficiency and we demonstrate this on high resolution meshes with optimizations involving over 26 million degrees of freedom. Making an existing solver differentiable requires only a few modifications and the model is compatible with both modern CPU and GPU multi-core hardware.
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.
Graph Convolution Networks (GCNs) manifest great potential in recommendation. This is attributed to their capability on learning good user and item embeddings by exploiting the collaborative signals from the high-order neighbors. Like other GCN models, the GCN based recommendation models also suffer from the notorious over-smoothing problem - when stacking more layers, node embeddings become more similar and eventually indistinguishable, resulted in performance degradation. The recently proposed LightGCN and LR-GCN alleviate this problem to some extent, however, we argue that they overlook an important factor for the over-smoothing problem in recommendation, that is, high-order neighboring users with no common interests of a user can be also involved in the user's embedding learning in the graph convolution operation. As a result, the multi-layer graph convolution will make users with dissimilar interests have similar embeddings. In this paper, we propose a novel Interest-aware Message-Passing GCN (IMP-GCN) recommendation model, which performs high-order graph convolution inside subgraphs. The subgraph consists of users with similar interests and their interacted items. To form the subgraphs, we design an unsupervised subgraph generation module, which can effectively identify users with common interests by exploiting both user feature and graph structure. To this end, our model can avoid propagating negative information from high-order neighbors into embedding learning. Experimental results on three large-scale benchmark datasets show that our model can gain performance improvement by stacking more layers and outperform the state-of-the-art GCN-based recommendation models significantly.
Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
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