Scientific machine learning has been successfully applied to inverse problems and PDE discoveries in computational physics. One caveat of current methods however is the need for large amounts of (clean) data in order to recover full system responses or underlying physical models. Bayesian methods may be particularly promising to overcome these challenges as they are naturally less sensitive to sparse and noisy data. In this paper, we propose to use Bayesian neural networks (BNN) in order to: 1) Recover the full system states from measurement data (e.g. temperature, velocity field, etc.). We use Hamiltonian Monte-Carlo to sample the posterior distribution of a deep and dense BNN, and show that it is possible to accurately capture physics of varying complexity without overfitting. 2) Recover the parameters in the underlying partial differential equation (PDE) governing the physical system. Using the trained BNN as a surrogate of the system response, we generate datasets of derivatives potentially comprising the latent PDE of the observed system and perform a Bayesian linear regression (BLR) between the successive derivatives in space and time to recover the original PDE parameters. We take advantage of the confidence intervals on the BNN outputs and introduce the spatial derivative variance into the BLR likelihood to discard the influence of highly uncertain surrogate data points, which allows for more accurate parameter discovery. We demonstrate our approach on a handful of example applied to physics and non-linear dynamics.
相關內容
Deep neural networks tend to underestimate uncertainty and produce overly confident predictions. Recently proposed solutions, such as MC Dropout and SDENet, require complex training and/or auxiliary out-of-distribution data. We propose a simple solution by extending the time-tested iterative reweighted least square (IRLS) in generalised linear regression. We use two sub-networks to parametrise the prediction and uncertainty estimation, enabling easy handling of complex inputs and nonlinear response. The two sub-networks have shared representations and are trained via two complementary loss functions for the prediction and the uncertainty estimates, with interleaving steps as in a cooperative game. Compared with more complex models such as MC-Dropout or SDE-Net, our proposed network is simpler to implement and more robust (insensitive to varying aleatoric and epistemic uncertainty).
We study the frontier between learnable and unlearnable hidden Markov models (HMMs). HMMs are flexible tools for clustering dependent data coming from unknown populations. The model parameters are known to be fully identifiable (up to label-switching) without any modeling assumption on the distributions of the populations as soon as the clusters are distinct and the hidden chain is ergodic with a full rank transition matrix. In the limit as any one of these conditions fails, it becomes impossible in general to identify parameters. For a chain with two hidden states we prove nonasymptotic minimax upper and lower bounds, matching up to constants, which exhibit thresholds at which the parameters become learnable. We also provide an upper bound on the relative entropy rate for parameters in a neighbourhood of the unlearnable region which may have interest in itself.
Bayesian learning via Stochastic Gradient Langevin Dynamics (SGLD) has been suggested for differentially private learning. While previous research provides differential privacy bounds for SGLD when close to convergence or at the initial steps of the algorithm, the question of what differential privacy guarantees can be made in between remains unanswered. This interim region is essential, especially for Bayesian neural networks, as it is hard to guarantee convergence to the posterior. This paper will show that using SGLD might result in unbounded privacy loss for this interim region, even when sampling from the posterior is as differentially private as desired.
Bayesian learning via Stochastic Gradient Langevin Dynamics (SGLD) has been suggested for differentially private learning. While previous research provides differential privacy bounds for SGLD when close to convergence or at the initial steps of the algorithm, the question of what differential privacy guarantees can be made in between remains unanswered. This interim region is essential, especially for Bayesian neural networks, as it is hard to guarantee convergence to the posterior. This paper will show that using SGLD might result in unbounded privacy loss for this interim region, even when sampling from the posterior is as differentially private as desired.
Solving for detailed chemical kinetics remains one of the major bottlenecks for computational fluid dynamics simulations of reacting flows using a finite-rate-chemistry approach. This has motivated the use of fully connected artificial neural networks to predict stiff chemical source terms as functions of the thermochemical state of the combustion system. However, due to the nonlinearities and multi-scale nature of combustion, the predicted solution often diverges from the true solution when these deep learning models are coupled with a computational fluid dynamics solver. This is because these approaches minimize the error during training without guaranteeing successful integration with ordinary differential equation solvers. In the present work, a novel neural ordinary differential equations approach to modeling chemical kinetics, termed as ChemNODE, is developed. In this deep learning framework, the chemical source terms predicted by the neural networks are integrated during training, and by computing the required derivatives, the neural network weights are adjusted accordingly to minimize the difference between the predicted and ground-truth solution. A proof-of-concept study is performed with ChemNODE for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions. It is shown that ChemNODE accurately captures the correct physical behavior and reproduces the results obtained using the full chemical kinetic mechanism at a fraction of the computational cost.
Data-driven methods are becoming an essential part of computational mechanics due to their unique advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form approximations. However, imposing the physics-based mathematical requirements that any material model must comply with is not straightforward for data-driven approaches. In this study, we use a novel class of neural networks, known as neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy function with respect to the deformation gradient, a condition needed for the existence of minimizers for boundary value problems in elasticity. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy function with respect to the invariants of the right Cauchy-Green deformation tensor. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations. The framework is general and can be used to model a large class of materials. Here we focus on hyperelasticity, but polyconvex strain energies are a core building block for other problems in elasticity such as viscous and plastic deformations. We therefore expect our methodology to further enable data-driven methods in computational mechanics
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.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191