Large-scale numerical simulations are used across many scientific disciplines to facilitate experimental development and provide insights into underlying physical processes, but they come with a significant computational cost. Deep neural networks (DNNs) can serve as highly-accurate surrogate models, with the capacity to handle diverse datatypes, offering tremendous speed-ups for prediction and many other downstream tasks. An important use-case for these surrogates is the comparison between simulations and experiments; prediction uncertainty estimates are crucial for making such comparisons meaningful, yet standard DNNs do not provide them. In this work we define the fundamental requirements for a DNN to be useful for scientific applications, and demonstrate a general variational inference approach to equip predictions of scalar and image data from a DNN surrogate model trained on inertial confinement fusion simulations with calibrated Bayesian uncertainties. Critically, these uncertainties are interpretable, meaningful and preserve physics-correlations in the predicted quantities.

Models of discrete-valued outcomes are easily misspecified if the data exhibit zero-inflation, overdispersion or contamination. Without additional knowledge about the existence and nature of this misspecification, model inference and prediction are adversely affected. Here, we introduce a robust discrepancy-based Bayesian approach using the Total Variation Distance (TVD). In the process, we address and resolve two challenges: First, we study convergence and robustness properties of a computationally efficient estimator for the TVD between a parametric model and the data-generating mechanism. Second, we provide an efficient inference method adapted from Lyddon et al. (2019) which corresponds to formulating an uninformative nonparametric prior directly over the data-generating mechanism. Lastly, we empirically demonstrate that our approach is robust and significantly improves predictive performance on a range of simulated and real world data.

Purpose: Bayesian calibration is theoretically superior to standard direct-search algorithm because it can reveal the full joint posterior distribution of the calibrated parameters. However, to date, Bayesian calibration has not been used often in health decision sciences due to practical and computational burdens. In this paper we propose to use artificial neural networks (ANN) as one solution to these limitations. Methods: Bayesian Calibration using Artificial Neural Networks (BayCANN) involves (1) training an ANN metamodel on a sample of model inputs and outputs, and (2) then calibrating the trained ANN metamodel instead of the full model in a probabilistic programming language to obtain the posterior joint distribution of the calibrated parameters. We demonstrate BayCANN by calibrating a natural history model of colorectal cancer to adenoma prevalence and cancer incidence data. In addition, we compare the efficiency and accuracy of BayCANN against performing a Bayesian calibration directly on the simulation model using an incremental mixture importance sampling (IMIS) algorithm. Results: BayCANN was generally more accurate than IMIS in recovering the "true" parameter values. The ratio of the absolute ANN deviation from the truth compared to IMIS for eight out of the nine calibrated parameters were less than one indicating that BayCANN was more accurate than IMIS. In addition, BayCANN took about 15 minutes total compared to the IMIS method which took 80 minutes. Conclusions: In our case study, BayCANN was more accurate than IMIS and was five-folds faster. Because BayCANN does not depend on the structure of the simulation model, it can be adapted to models of various levels of complexity with minor changes to its structure. We provide BayCANN's open-source implementation in R.

Decision making for autonomous driving in urban environments is challenging due to the complexity of the road structure and the uncertainty in the behavior of diverse road users. Traditional methods consist of manually designed rules as the driving policy, which require expert domain knowledge, are difficult to generalize and might give sub-optimal results as the environment gets complex. Whereas, using reinforcement learning, optimal driving policy could be learned and improved automatically through several interactions with the environment. However, current research in the field of reinforcement learning for autonomous driving is mainly focused on highway setup with little to no emphasis on urban environments. In this work, a deep reinforcement learning based decision-making approach for high-level driving behavior is proposed for urban environments in the presence of pedestrians. For this, the use of Deep Recurrent Q-Network (DRQN) is explored, a method combining state-of-the art Deep Q-Network (DQN) with a long term short term memory (LSTM) layer helping the agent gain a memory of the environment. A 3-D state representation is designed as the input combined with a well defined reward function to train the agent for learning an appropriate behavior policy in a real-world like urban simulator. The proposed method is evaluated for dense urban scenarios and compared with a rule-based approach and results show that the proposed DRQN based driving behavior decision maker outperforms the rule-based approach.

Many processes, such as discrete event systems in engineering or population dynamics in biology, evolve in discrete space and continuous time. We consider the problem of optimal decision making in such discrete state and action space systems under partial observability. This places our work at the intersection of optimal filtering and optimal control. At the current state of research, a mathematical description for simultaneous decision making and filtering in continuous time with finite state and action spaces is still missing. In this paper, we give a mathematical description of a continuous-time partial observable Markov decision process (POMDP). By leveraging optimal filtering theory we derive a Hamilton-Jacobi-Bellman (HJB) type equation that characterizes the optimal solution. Using techniques from deep learning we approximately solve the resulting partial integro-differential equation. We present (i) an approach solving the decision problem offline by learning an approximation of the value function and (ii) an online algorithm which provides a solution in belief space using deep reinforcement learning. We show the applicability on a set of toy examples which pave the way for future methods providing solutions for high dimensional problems.

Federated learning aims to collaboratively train a strong global model by accessing users' locally trained models but not their own data. A crucial step is therefore to aggregate local models into a global model, which has been shown challenging when users have non-i.i.d. data. In this paper, we propose a novel aggregation algorithm named FedBE, which takes a Bayesian inference perspective by sampling higher-quality global models and combining them via Bayesian model Ensemble, leading to much robust aggregation. We show that an effective model distribution can be constructed by simply fitting a Gaussian or Dirichlet distribution to the local models. Our empirical studies validate FedBE's superior performance, especially when users' data are not i.i.d. and when the neural networks go deeper. Moreover, FedBE is compatible with recent efforts in regularizing users' model training, making it an easily applicable module: you only need to replace the aggregation method but leave other parts of your federated learning algorithm intact.

We introduce a new theoretical framework to analyze deep learning optimization with connection to its generalization error. Existing frameworks such as mean field theory and neural tangent kernel theory for neural network optimization analysis typically require taking limit of infinite width of the network to show its global convergence. This potentially makes it difficult to directly deal with finite width network; especially in the neural tangent kernel regime, we cannot reveal favorable properties of neural networks beyond kernel methods. To realize more natural analysis, we consider a completely different approach in which we formulate the parameter training as a transportation map estimation and show its global convergence via the theory of the infinite dimensional Langevin dynamics. This enables us to analyze narrow and wide networks in a unifying manner. Moreover, we give generalization gap and excess risk bounds for the solution obtained by the dynamics. The excess risk bound achieves the so-called fast learning rate. In particular, we show an exponential convergence for a classification problem and a minimax optimal rate for a regression problem.

Learning expressive probabilistic models correctly describing the data is a ubiquitous problem in machine learning. A popular approach for solving it is mapping the observations into a representation space with a simple joint distribution, which can typically be written as a product of its marginals -- thus drawing a connection with the field of nonlinear independent component analysis. Deep density models have been widely used for this task, but their maximum likelihood based training requires estimating the log-determinant of the Jacobian and is computationally expensive, thus imposing a trade-off between computation and expressive power. In this work, we propose a new approach for exact training of such neural networks. Based on relative gradients, we exploit the matrix structure of neural network parameters to compute updates efficiently even in high-dimensional spaces; the computational cost of the training is quadratic in the input size, in contrast with the cubic scaling of naive approaches. This allows fast training with objective functions involving the log-determinant of the Jacobian, without imposing constraints on its structure, in stark contrast to autoregressive normalizing flows.

The dominant term in PAC-Bayes bounds is often the Kullback--Leibler divergence between the posterior and prior. For so-called linear PAC-Bayes risk bounds based on the empirical risk of a fixed posterior kernel, it is possible to minimize the expected value of the bound by choosing the prior to be the expected posterior, which we call the oracle prior on the account that it is distribution dependent. In this work, we show that the bound based on the oracle prior can be suboptimal: In some cases, a stronger bound is obtained by using a data-dependent oracle prior, i.e., a conditional expectation of the posterior, given a subset of the training data that is then excluded from the empirical risk term. While using data to learn a prior is a known heuristic, its essential role in optimal bounds is new. In fact, we show that using data can mean the difference between vacuous and nonvacuous bounds. We apply this new principle in the setting of nonconvex learning, simulating data-dependent oracle priors on MNIST and Fashion MNIST with and without held-out data, and demonstrating new nonvacuous bounds in both cases.

In the following, we assume to have access to a differential equation system Neural Stochastic Differential Equations that describes the dynamics of the target environment model a dynamical environment with neural with low fidelity, e.g. by describing the vector field on nets assigned to their drift and diffusion a reduced dimensionality, by ignoring detailed models terms. The high expressive power of their of some system components, or by avoiding certain nonlinearity comes at the expense of instability dependencies for computational feasibility. We incorporate in the identification of the large set of free the ODE system provided by the domain expert parameters. This paper presents a recipe to into a nonlinear system identification engine, which we improve the prediction accuracy of such models choose to be a Bayesian Neural Stochastic Differential in three steps: i) accounting for epistemic Equation (BNSDE) to cover a large scope of dynamical uncertainty by assuming probabilistic weights, systems, resulting in a hybrid model.