The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g. civil or mechanical structures), which are typically high-dimensional in nature. In the scope of physics-informed machine learning, this paper proposes a framework - termed Neural Modal ODEs - to integrate physics-based modeling with deep learning (particularly, Neural Ordinary Differential Equations -- Neural ODEs) for modeling the dynamics of monitored and high-dimensional engineered systems. In this initiating exploration, we restrict ourselves to linear or mildly nonlinear systems. We propose an architecture that couples a dynamic version of variational autoencoders with physics-informed Neural ODEs (Pi-Neural ODEs). An encoder, as a part of the autoencoder, learns the abstract mappings from the first few items of observational data to the initial values of the latent variables, which drive the learning of embedded dynamics via physics-informed Neural ODEs, imposing a \textit{modal model} structure to that latent space. The decoder of the proposed model adopts the eigenmodes derived from an eigen-analysis applied to the linearized portion of a physics-based model: a process implicitly carrying the spatial relationship between degrees-of-freedom (DOFs). The framework is validated on a numerical example, and an experimental dataset of a scaled cable-stayed bridge, where the learned hybrid model is shown to outperform a purely physics-based approach to modeling. We further show the functionality of the proposed scheme within the context of virtual sensing, i.e., the recovery of generalized response quantities in unmeasured DOFs from spatially sparse data.
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Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis (TDA) is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyse some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 Bridge case study in structural health monitoring, where it will be used to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.
Due to inevitable noises introduced during scanning and quantization, 3D reconstruction via RGB-D sensors suffers from errors both in geometry and texture, leading to artifacts such as camera drifting, mesh distortion, texture ghosting, and blurriness. Given an imperfect reconstructed 3D model, most previous methods have focused on the refinement of either geometry, texture, or camera pose. Or different optimization schemes and objectives for optimizing each component have been used in previous joint optimization methods, forming a complicated system. In this paper, we propose a novel optimization approach based on differentiable rendering, which integrates the optimization of camera pose, geometry, and texture into a unified framework by enforcing consistency between the rendered results and the corresponding RGB-D inputs. Based on the unified framework, we introduce a joint optimization approach to fully exploit the inter-relationships between geometry, texture, and camera pose, and describe an adaptive interleaving strategy to improve optimization stability and efficiency. Using differentiable rendering, an image-level adversarial loss is applied to further improve the 3D model, making it more photorealistic. Experiments on synthetic and real data using quantitative and qualitative evaluation demonstrated the superiority of our approach in recovering both fine-scale geometry and high-fidelity texture.Code is available at //adjointopti.github.io/adjoin.github.io/.
In this paper, we propose two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (PDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully nonlinear PDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully nonlinear PDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multi-scale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. In numerical experiments, involving Allen-Cahn equations, Black-Scholes-Barentblatt equations, and Hamiltonian-Jacobi-Bellman equations, the first proposed method exhibits higher efficiency and accuracy than the existing method; the second proposed method can extend the dimensionality of the completely non-linear PDEs-2BSDEs system over $400$ dimensions, from which the numerical results illustrate the effectiveness of proposed methods.
A key challenge for a common waveform for Integrated Sensing and Communications (ISAC) - widely seen as an attractive proposition to achieve high performance for both functionalities, while efficiently utilizing available resources -- lies in leveraging information-bearing channel-coded communications signals (c.c.s) for sensing. In this paper, we investigate the sensing performance of c.c.s in (multi-user) interference-limited operation, and show that it is limited by sidelobes in the range-Doppler map, whose form depends on whether the c.c.s modulates a single-carrier or OFDM waveform. While uncoded communications signals -- comprising a block of $N$ i.i.d zero-mean symbols -- give rise to asymptotically (i.e., as $N \rightarrow \infty$) zero sidelobes due to the law of large numbers, it is not obvious that the same holds for c.c.s, as structured channel coding schemes (e.g., linear block codes) induce dependence across codeword symbols. In this paper, we show that c.c.s also give rise to asymptotically zero sidelobes -- for both single-carrier and OFDM waveforms -- by deriving upper bounds for the tail probabilities of the sidelobe magnitudes that decay as $\exp( - O($code rate $\times$ block length$))$. This implies that for any code rate, c.c.s are effective sensing signals that are robust to multi-user interference at sufficiently large block lengths, with negligible difference in performance based on whether they modulate a single-carrier or OFDM waveform. We verify the latter implication through simulations, where we observe the sensing performance (characterized by the detection and false-alarm probabilities) of a QPSK-modulated c.c.s (code rate = 120/1024, block length = 1024 symbols) to match that of a comparable interference-free FMCW waveform even at high interference levels (signal-to-interference ratio of -11dB), for both single-carrier and OFDM waveforms.
Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. Despite recent progress on expanding the class of problems that can be tackled by PINNs, most of existing use-cases involve simple geometric domains. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. In this work, we propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator. This technique allows to create an input space for the neural network that represents the geometry of a given object. We approximate the eigenfunctions as well as the operators involved in the partial differential equations with finite elements. We extensively test and compare the proposed methodology against traditional PINNs in complex shapes, such as a coil, a heat sink and a bunny, with different physics, such as the Eikonal equation and heat transfer. We also study the sensitivity of our method to the number of eigenfunctions used, as well as the discretization used for the eigenfunctions and the underlying operators. Our results show excellent agreement with the ground truth data in cases where traditional PINNs fail to produce a meaningful solution. We envision this new technique will expand the effectiveness of PINNs to more realistic applications.
The term NeuralODE describes the structural combination of an Artifical Neural Network (ANN) and a numerical solver for Ordinary Differential Equations (ODEs), the former acts as the right-hand side of the ODE to be solved. This concept was further extended by a black-box model in the form of a Functional Mock-up Unit (FMU) to obtain a subclass of NeuralODEs, named NeuralFMUs. The resulting structure features the advantages of first-principle and data-driven modeling approaches in one single simulation model: A higher prediction accuracy compared to conventional First Principle Models (FPMs), while also a lower training effort compared to purely data-driven models. We present an intuitive workflow to setup and use NeuralFMUs, enabling the encapsulation and reuse of existing conventional models exported from common modeling tools. Moreover, we exemplify this concept by deploying a NeuralFMU for a consumption simulation based on a Vehicle Longitudinal Dynamics Model (VLDM), which is a typical use case in automotive industry. Related challenges that are often neglected in scientific use cases, like real measurements (e.g. noise), an unknown system state or high-frequent discontinuities, are handled in this contribution. For the aim to build a hybrid model with a higher prediction quality than the original FPM, we briefly highlight two open-source libraries: FMI.jl for integrating FMUs into the Julia programming environment, as well as an extension to this library called FMIFlux.jl, that allows for the integration of FMUs into a neural network topology to finally obtain a NeuralFMU.
We consider the neural ODE and optimal control perspective of supervised learning, with $\ell^1$-control penalties, where rather than only minimizing a final cost (the \emph{empirical risk}) for the state, we integrate this cost over the entire time horizon. We prove that any optimal control (for this cost) vanishes beyond some positive stopping time. When seen in the discrete-time context, this result entails an \emph{ordered} sparsity pattern for the parameters of the associated residual neural network: ordered in the sense that these parameters are all $0$ beyond a certain layer. Furthermore, we provide a polynomial stability estimate for the empirical risk with respect to the time horizon. This can be seen as a \emph{turnpike property}, for nonsmooth dynamics and functionals with $\ell^1$-penalties, and without any smallness assumptions on the data, both of which are new in the literature.
A fundamental property of deep learning normalization techniques, such as batch normalization, is making the pre-normalization parameters scale invariant. The intrinsic domain of such parameters is the unit sphere, and therefore their gradient optimization dynamics can be represented via spherical optimization with varying effective learning rate (ELR), which was studied previously. In this work, we investigate the properties of training scale-invariant neural networks directly on the sphere using a fixed ELR. We discover three regimes of such training depending on the ELR value: convergence, chaotic equilibrium, and divergence. We study these regimes in detail both on a theoretical examination of a toy example and on a thorough empirical analysis of real scale-invariant deep learning models. Each regime has unique features and reflects specific properties of the intrinsic loss landscape, some of which have strong parallels with previous research on both regular and scale-invariant neural networks training. Finally, we demonstrate how the discovered regimes are reflected in conventional training of normalized networks and how they can be leveraged to achieve better optima.
Graph Neural Networks (GNNs) have recently become increasingly popular due to their ability to learn complex systems of relations or interactions arising in a broad spectrum of problems ranging from biology and particle physics to social networks and recommendation systems. Despite the plethora of different models for deep learning on graphs, few approaches have been proposed thus far for dealing with graphs that present some sort of dynamic nature (e.g. evolving features or connectivity over time). In this paper, we present Temporal Graph Networks (TGNs), a generic, efficient framework for deep learning on dynamic graphs represented as sequences of timed events. Thanks to a novel combination of memory modules and graph-based operators, TGNs are able to significantly outperform previous approaches being at the same time more computationally efficient. We furthermore show that several previous models for learning on dynamic graphs can be cast as specific instances of our framework. We perform a detailed ablation study of different components of our framework and devise the best configuration that achieves state-of-the-art performance on several transductive and inductive prediction tasks for dynamic graphs.
Most object recognition approaches predominantly focus on learning discriminative visual patterns while overlooking the holistic object structure. Though important, structure modeling usually requires significant manual annotations and therefore is labor-intensive. In this paper, we propose to "look into object" (explicitly yet intrinsically model the object structure) through incorporating self-supervisions into the traditional framework. We show the recognition backbone can be substantially enhanced for more robust representation learning, without any cost of extra annotation and inference speed. Specifically, we first propose an object-extent learning module for localizing the object according to the visual patterns shared among the instances in the same category. We then design a spatial context learning module for modeling the internal structures of the object, through predicting the relative positions within the extent. These two modules can be easily plugged into any backbone networks during training and detached at inference time. Extensive experiments show that our look-into-object approach (LIO) achieves large performance gain on a number of benchmarks, including generic object recognition (ImageNet) and fine-grained object recognition tasks (CUB, Cars, Aircraft). We also show that this learning paradigm is highly generalizable to other tasks such as object detection and segmentation (MS COCO). Project page: //github.com/JDAI-CV/LIO.
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