Analyzing Inverse Problems with Invertible Neural Networks
Visual Learning Lab Heidelberg（海德堡视觉学习实验室）；German Cancer Research Center (DKFZ)（德国癌症研究中心（DKFZ））；Zentrum für Astronomie der Universität Heidelberg (ZAH)（海德堡大学天文中心（ZAH））
Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert, Daniel Rahner, Eric W. Pellegrini, Ralf S. Klessen, Lena Maier-Hein, Carsten Rother, Ullrich Köthe
In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution, conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task -- so-called Invertible Neural Networks (INNs). Although INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.
University of Bremen（不来梅大学）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Invertible neural networks (INNs) have been used to design generative models, implement memory-saving gradient computation, and solve inverse problems. In this work, we show that commonly-used INN architectures suffer from exploding inverses and are thus prone to becoming numerically non-invertible. Across a wide range of INN use-cases, we reveal failures including the non-applicability of the change-of-variables formula on in- and out-of-distribution (OOD) data, incorrect gradients for memory-saving backprop, and the inability to sample from normalizing flow models. We further derive bi-Lipsc…
Recent work demonstrated that flow-based invertible neural networks are promising tools for solving ambiguous inverse problems. Following up on this, we investigate how ten invertible architectures and related models fare on two intuitive, low-dimensional benchmark problems, obtaining the best results with coupling layers and simple autoencoders. We hope that our initial efforts inspire other researchers to evaluate their invertible architectures in the same setting and put forth additional benchmarks, so our evaluation may eventually grow into an official community challenge.
Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine transformations. The significant advantage of such models is their easy-to-compute inverse. Nevertheless, making use of affine transformations may limit the expressiveness of such models. Recently, invertible piecewise polynomial functions as a replacement for affine transformati…
An invertible generative model for forward and inverse problems
Centrum Wiskunde en Informatica, Amsterdam, The Netherlands（荷兰阿姆斯特丹数学与信息学研究中心）；Utrecht University, Utrecht, The Netherlands（荷兰乌得勒支大学）；University of Twente, Enschede, The Netherlands（荷兰埃因霍温大学）
Technical University of Munich（慕尼黑工业大学）；University of Pennsylvania（宾夕法尼亚大学）；Munich Data Science Institute（慕尼黑数据科学研究所）
Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and diff…
TU Berlin（柏林工业大学）；Physikalisch-Technische Bundesanstalt（德国联邦物理技术研究院）
In this paper, we analyze the properties of invertible neural networks, which provide a way of solving inverse problems. Our main focus lies on investigating and controlling the Lipschitz constants of the corresponding inverse networks. Without such an control, numerical simulations are prone to errors and not much is gained against traditional approaches. Fortunately, our analysis indicates that changing the latent distribution from a standard normal one to a Gaussian mixture model resolves the issue of exploding Lipschitz constants. Indeed, numerical simulations confirm that this modificatio…
Inverse design, the process of matching a device or process parameters to exhibit a desired performance, is applied in many disciplines ranging from material design over chemical processes and to engineering. Machine learning has emerged as a promising approach to overcome current limitations imposed by the dimensionality of the parameter space and multimodal parameter distributions. Most traditional optimization routines assume an invertible one-to-one mapping between the design parameters and the target performance. However, comparable or even identical performance may be realized by differe…
Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for …
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
Graph prediction problems prevail in data analysis and machine learning. The inverse prediction problem, namely to infer input data from given output labels, is of emerging interest in various applications. In this work, we develop \textit{invertible graph neural network} (iGNN), a deep generative model to tackle the inverse prediction problem on graphs by casting it as a conditional generative task. The proposed model consists of an invertible sub-network that maps one-to-one from data to an intermediate encoded feature, which allows forward prediction by a linear classification sub-network a…
Most problems in natural language processing can be approximated as inverse problems such as analysis and generation at variety of levels from morphological (e.g., cat+Plural <-> cats) to semantic (e.g., (call + 1 2) <-> "Calculate one plus two."). Although the tasks in both directions are closely related, general approach in the field has been to design separate models specific for each task. However, having one shared model for both tasks, would help the researchers exploit the common knowledge among these problems with reduced time and memory requirements. We investigate a specific class of…
Oak Ridge National Laboratory（橡树岭国家实验室）；Queens College of the City University of New York（纽约市立大学皇后学院）
The ability to readily design novel materials with chosen functional properties on-demand represents a next frontier in materials discovery. However, thoroughly and efficiently sampling the entire design space in a computationally tractable manner remains a highly challenging task. To tackle this problem, we propose an inverse design framework (MatDesINNe) utilizing invertible neural networks which can map both forward and reverse processes between the design space and target property. This approach can be used to generate materials candidates for a designated property, thereby satisfying the …
The encoder-decoder models for unsupervised sentence representation learning tend to discard the decoder after being trained on a large unlabelled corpus, since only the encoder is needed to map the input sentence into a vector representation. However, parameters learnt in the decoder also contain useful information about language. In order to utilise the decoder after learning, we present two types of decoding functions whose inverse can be easily derived without expensive inverse calculation. Therefore, the inverse of the decoding function serves as another encoder that produces sentence rep…
Xiaomi AI Lab（小米人工智能实验室）；The Chinese University of Hong Kong（香港中文大学）
Generative Adversarial Networks (GANs) play an increasingly important role in machine learning. However, there is one fundamental issue hindering their practical applications: the absence of capability for encoding real-world samples. The conventional way of addressing this issue is to learn an encoder for GAN via Variational Auto-Encoder (VAE). In this paper, we show that the entanglement of the latent space for the VAE/GAN framework poses the main challenge for encoder learning. To address the entanglement issue and enable inference in GAN we propose a novel algorithm named Latently Invertib…
Singapore-MIT Alliance for Research and Technology（新加坡-麻省理工学院研究与技术联盟）；National University of Singapore（新加坡国立大学）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Massachusetts Institute of Technology（麻省理工学院）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Agency for Science, Technology and Research（新加坡科技研究局）；Nanyang Technological University（南洋理工大学）；Shanghai University（上海大学）；Microsoft AI for Good（微软人工智能公益计划）；Toyota Research Institute（丰田研究院）；Tsinghua University（清华大学）
Realizing general inverse design could greatly accelerate the discovery of new materials with user-defined properties. However, state-of-the-art generative models tend to be limited to a specific composition or crystal structure. Herein, we present a framework capable of general inverse design (not limited to a given set of elements or crystal structures), featuring a generalized invertible representation that encodes crystals in both real and reciprocal space, and a property-structured latent space from a variational autoencoder (VAE). In three design cases, the framework generates 142 new cr…
In this paper we analyse and improve integer discrete flows for lossless compression. Integer discrete flows are a recently proposed class of models that learn invertible transformations for integer-valued random variables. Their discrete nature makes them particularly suitable for lossless compression with entropy coding schemes. We start by investigating a recent theoretical claim that states that invertible flows for discrete random variables are less flexible than their continuous counterparts. We demonstrate with a proof that this claim does not hold for integer discrete flows due to the …
Massachusetts Institute of Technology（麻省理工学院）；Scripps Institution of Oceanography（斯克里普斯海洋研究所）；Paul Scherrer Institute（保罗·谢勒研究所）；University of California San Diego（加利福尼亚大学圣迭戈分校）
In this work, we investigate the use of normalizing flows to model conditional distributions. In particular, we use our proposed method to analyze inverse problems with invertible neural networks by maximizing the posterior likelihood. Our method uses only a single loss and is easy to train. This is an improvement on the previous method that solves similar inverse problems with invertible neural networks but which involves a combination of several loss terms with ad-hoc weighting. In addition, our method provides a natural framework to incorporate conditioning in normalizing flows, and therefo…
Physikalisch-Technische Bundesanstalt Braunschweig und Berlin（德国联邦物理技术研究院（不伦瑞克-柏林））；TU Berlin（柏林工业大学）
Grazing incidence X-ray fluorescence is a non-destructive technique for analyzing the geometry and compositional parameters of nanostructures appearing e.g. in computer chips. In this paper, we propose to reconstruct the posterior parameter distribution given a noisy measurement generated by the forward model by an appropriately learned invertible neural network. This network resembles the transport map from a reference distribution to the posterior. We demonstrate by numerical comparisons that our method can compete with established Markov Chain Monte Carlo approaches, while being more effici…
We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior pe…
U-Nets have been established as a standard architecture for image-to-image learning problems such as segmentation and inverse problems in imaging. For large-scale data, as it for example appears in 3D medical imaging, the U-Net however has prohibitive memory requirements. Here, we present a new fully-invertible U-Net-based architecture called the iUNet, which employs novel learnable and invertible up- and downsampling operations, thereby making the use of memory-efficient backpropagation possible. This allows us to train deeper and larger networks in practice, under the same GPU memory restric…
Freie Universität Berlin（柏林自由大学）；Rice University（莱斯大学）
Many types of neural network layers rely on matrix properties such as invertibility or orthogonality. Retaining such properties during optimization with gradient-based stochastic optimizers is a challenging task, which is usually addressed by either reparameterization of the affected parameters or by directly optimizing on the manifold. This work presents a novel approach for training invertible linear layers. In lieu of directly optimizing the network parameters, we train rank-one perturbations and add them to the actual weight matrices infrequently. This P$^{4}$Inv update allows keeping trac…
Peking University（北京大学）；Microsoft Research Asia（微软亚洲研究院）；Peng Cheng Laboratory（鹏城实验室）
Image rescaling is a commonly used bidirectional operation, which first downscales high-resolution images to fit various display screens or to be storage- and bandwidth-friendly, and afterward upscales the corresponding low-resolution images to recover the original resolution or the details in the zoom-in images. However, the non-injective downscaling mapping discards high-frequency contents, leading to the ill-posed problem for the inverse restoration task. This can be abstracted as a general image degradation-restoration problem with information loss. In this work, we propose a novel inverti…
Localizing the source of graph diffusion phenomena, such as misinformation propagation, is an important yet extremely challenging task. Existing source localization models typically are heavily dependent on the hand-crafted rules. Unfortunately, a large portion of the graph diffusion process for many applications is still unknown to human beings so it is important to have expressive models for learning such underlying rules automatically. This paper aims to establish a generic framework of invertible graph diffusion models for source localization on graphs, namely Invertible Validity-aware Gra…
Understanding Neural Network Systems for Image Analysis using Vector Spaces and Inverse Maps
Department of Computer Science（计算机科学系）；The University of California at Los Angeles（加州大学洛杉矶分校）；Department of Electrical and Computer Engineering（电气与计算机工程系）；The University of New Mexico, USA（新墨西哥大学）
Deep learning (DL) inverse techniques have increased the speed of artificial electromagnetic material (AEM) design and improved the quality of resulting devices. Many DL inverse techniques have succeeded on a number of AEM design tasks, but to compare, contrast, and evaluate assorted techniques it is critical to clarify the underlying ill-posedness of inverse problems. Here we review state-of-the-art approaches and present a comprehensive survey of deep learning inverse methods and invertible and conditional invertible neural networks to AEM design. We produce easily accessible and rapidly imp…
A method for nonlinear topology identification is proposed, based on the assumption that a collection of time series are generated in two steps: i) a vector autoregressive process in a latent space, and ii) a nonlinear, component-wise, monotonically increasing observation mapping. The latter mappings are assumed invertible, and are modelled as shallow neural networks, so that their inverse can be numerically evaluated, and their parameters can be learned using a technique inspired in deep learning. Due to the function inversion, the back-propagation step is not straightforward, and this paper …
Intuit AI Research（Intuit AI研究院）；Walmart Global Tech（沃尔玛全球技术）；Georgia Institute of Technology（佐治亚理工学院）
In the scope of "AI for Science", solving inverse problems is a longstanding challenge in materials and drug discovery, where the goal is to determine the hidden structures given a set of desirable properties. Deep generative models are recently proposed to solve inverse problems, but these currently use expensive forward operators and struggle in precisely localizing the exact solutions and fully exploring the parameter spaces without missing solutions. In this work, we propose a novel approach (called iPage) to accelerate the inverse learning process by leveraging probabilistic inference fro…
