Model-based deep learning (MoDL) algorithms that rely on unrolling are emerging as powerful tools for image recovery. In this work, we introduce a novel monotone operator learning framework to overcome some of the challenges associated with current unrolled frameworks, including high memory cost, lack of guarantees on robustness to perturbations, and low interpretability. Unlike current unrolled architectures that use finite number of iterations, we use the deep equilibrium (DEQ) framework to iterate the algorithm to convergence and to evaluate the gradient of the convolutional neural network blocks using Jacobian iterations. This approach significantly reduces the memory demand, facilitating the extension of MoDL algorithms to high dimensional problems. We constrain the CNN to be a monotone operator, which allows us to introduce algorithms with guaranteed convergence properties and robustness guarantees. We demonstrate the utility of the proposed scheme in the context of parallel MRI.
The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.
In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. Specifically, in PINO, we combine coarse-resolution training data with PDE constraints imposed at a higher resolution. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families and shows no degradation in accuracy even under zero-shot super-resolution, i.e., being able to predict beyond the resolution of training data. PINO uses the Fourier neural operator (FNO) framework that is guaranteed to be a universal approximator for any continuous operator and discretization-convergent in the limit of mesh refinement. By adding PDE constraints to FNO at a higher resolution, we obtain a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO succeeds in settings where no training data is available and only PDE constraints are imposed, while previous approaches, such as the Physics-Informed Neural Network (PINN), fail due to optimization challenges, e.g., in multi-scale dynamic systems such as Kolmogorov flows.
Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a set of controllable variables through an experiment. In cases where the experimental dynamics can be described by partial differential equations (PDEs), such problems can be mathematically translated into PDE-constrained optimization tasks, which quickly become intractable as the number of control variables and the cost of experiments increases. In this work we leverage physics-informed deep operator networks (DeepONets) -- a self-supervised framework for learning the solution operator of parametric PDEs -- to build fast and differentiable surrogates for rapidly solving PDE-constrained optimization problems, even in the absence of any paired input-output training data. The effectiveness of the proposed framework will be demonstrated across different applications involving continuous functions as control or design variables, including time-dependent optimal control of heat transfer, and drag minimization of obstacles in Stokes flow. In all cases, we observe that DeepONets can minimize high-dimensional cost functionals in a matter of seconds, yielding a significant speed up compared to traditional adjoint PDE solvers that are typically costly and limited to relatively low-dimensional control/design parametrizations.
Operator learning techniques have recently emerged as a powerful tool for learning maps between infinite-dimensional Banach spaces. Trained under appropriate constraints, they can also be effective in learning the solution operator of partial differential equations (PDEs) in an entirely self-supervised manner. In this work we analyze the training dynamics of deep operator networks (DeepONets) through the lens of Neural Tangent Kernel (NTK) theory, and reveal a bias that favors the approximation of functions with larger magnitudes. To correct this bias we propose to adaptively re-weight the importance of each training example, and demonstrate how this procedure can effectively balance the magnitude of back-propagated gradients during training via gradient descent. We also propose a novel network architecture that is more resilient to vanishing gradient pathologies. Taken together, our developments provide new insights into the training of DeepONets and consistently improve their predictive accuracy by a factor of 10-50x, demonstrated in the challenging setting of learning PDE solution operators in the absence of paired input-output observations. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/ImprovedDeepONets.}
The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme} that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative $L2$ error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.
In this article, we perform a review of the state-of-the-art of hybrid machine learning in medical imaging. We start with a short summary of the general developments of the past in machine learning and how general and specialized approaches have been in competition in the past decades. A particular focus will be the theoretical and experimental evidence pro and contra hybrid modelling. Next, we inspect several new developments regarding hybrid machine learning with a particular focus on so-called known operator learning and how hybrid approaches gain more and more momentum across essentially all applications in medical imaging and medical image analysis. As we will point out by numerous examples, hybrid models are taking over in image reconstruction and analysis. Even domains such as physical simulation and scanner and acquisition design are being addressed using machine learning grey box modelling approaches. Towards the end of the article, we will investigate a few future directions and point out relevant areas in which hybrid modelling, meta learning, and other domains will likely be able to drive the state-of-the-art ahead.
In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.
Learning and solving governing equations of a physical system, represented by partial differential equations (PDEs), from data is a central challenge in a variety of areas of science and engineering. Traditional numerical methods for solving PDEs can be computationally expensive for complex systems and require the complete PDEs of the physical system. On the other hand, current data-driven machine learning methods require a large amount of data to learn a surrogate model of the PDE solution operator, which could be impractical. Here, we propose the first solution operator learning method that only requires one PDE solution, i.e., one-shot learning. By leveraging the principle of locality of PDEs, we consider small local domains instead of the entire computational domain and define a local solution operator. The local solution operator is then trained using a neural network, and utilized to predict the solution of a new input function via mesh-based fixed-point iteration (FPI), meshfree local-solution-operator informed neural network (LOINN) or local-solution-operator informed neural network with correction (cLOINN). We test our method on diverse PDEs, including linear or nonlinear PDEs, PDEs defined on complex geometries, and PDE systems, demonstrating the effectiveness and generalization capabilities of our method across these varied scenarios.
Robotic planning problems in hybrid state and action spaces can be solved by integrated task and motion planners (TAMP) that handle the complex interaction between motion-level decisions and task-level plan feasibility. TAMP approaches rely on domain-specific symbolic operators to guide the task-level search, making planning efficient. In this work, we formalize and study the problem of operator learning for TAMP. Central to this study is the view that operators define a lossy abstraction of the transition model of a domain. We then propose a bottom-up relational learning method for operator learning and show how the learned operators can be used for planning in a TAMP system. Experimentally, we provide results in three domains, including long-horizon robotic planning tasks. We find our approach to substantially outperform several baselines, including three graph neural network-based model-free approaches from the recent literature. Video: https://youtu.be/iVfpX9BpBRo Code: https://git.io/JCT0g
Image guidance in minimally invasive interventions is usually provided using live 2D X-ray imaging. To enhance the information available during the intervention, the preoperative volume can be overlaid over the 2D images using 2D/3D image registration. Recently, deep learning-based 2D/3D registration methods have shown promising results by improving computational efficiency and robustness. However, there is still a gap in terms of registration accuracy compared to traditional optimization-based methods. We aim to address this gap by incorporating traditional methods in deep neural networks using known operator learning. As an initial step in this direction, we propose to learn the update step of an iterative 2D/3D registration framework based on the Point-to-Plane Correspondence model. We embed the Point-to-Plane Correspondence model as a known operator in our deep neural network and learn the update step for the iterative registration. We show an improvement of 1.8 times in terms of registration accuracy for the update step prediction compared to learning without the known operator.
We study generalization under labeled shift for categorical and general normed label spaces. We propose a series of methods to estimate the importance weights from labeled source to unlabeled target domain and provide confidence bounds for these estimators. We deploy these estimators and provide generalization bounds in the unlabeled target domain.
The Tsetlin Machine (TM) is a recent machine learning algorithm with several distinct properties, such as interpretability, simplicity, and hardware-friendliness. Although numerous empirical evaluations report on its performance, the mathematical analysis of its convergence is still open. In this article, we analyze the convergence of the TM with only one clause involved for classification. More specifically, we examine two basic logical operators, namely, the "IDENTITY"- and "NOT" operators. Our analysis reveals that the TM, with just one clause, can converge correctly to the intended logical operator, learning from training data over an infinite time horizon. Besides, it can capture arbitrarily rare patterns and select the most accurate one when two candidate patterns are incompatible, by configuring a granularity parameter. The analysis of the convergence of the two basic operators lays the foundation for analyzing other logical operators. These analyses altogether, from a mathematical perspective, provide new insights on why TMs have obtained state-of-the-art performance on several pattern recognition problems.
We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(\frac{1}{\log T})$ for the constant and diminishing learning rates, respectively, under some mild conditions. To cope with the high dimensional nonlinear dynamical systems, we present the first-ever distributed Koopman operator learning algorithm. We show that the distributed Koopman operator learning has the same convergence properties as the centralized Koopman operator learning, in the absence of optimal tracker, so long as the basis functions satisfy a set of state-based decomposition conditions. Numerical experiments are provided to complement our theoretical results.
Convolutional analysis operator learning (CAOL) enables the unsupervised training of (hierarchical) convolutional sparsifying operators or autoencoders from large datasets. One can use many training images for CAOL, but a precise understanding of the impact of doing so has remained an open question. This paper presents a series of results that lend insight into the impact of dataset size on the filter update in CAOL. The first result is a general deterministic bound on errors in the estimated filters, and is followed by a bound on the expected errors as the number of training samples increases. The second result provides a high probability analogue. The bounds depend on properties of the training data, and we investigate their empirical values with real data. Taken together, these results provide evidence for the potential benefit of using more training data in CAOL.
Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be the sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator. In this work, we present an algorithm for learning an analysis operator from training images. Our method is based on an $\ell_p$-norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques.
This paper addresses a new learning algorithm for the recently introduced co-sparse analysis model. First, we give new insights into the co-sparse analysis model by establishing connections to filter-based MRF models, such as the Field of Experts (FoE) model of Roth and Black. For training, we introduce a technique called bi-level optimization to learn the analysis operators. Compared to existing analysis operator learning approaches, our training procedure has the advantage that it is unconstrained with respect to the analysis operator. We investigate the effect of different aspects of the co-sparse analysis model and show that the sparsity promoting function (also called penalty function) is the most important factor in the model. In order to demonstrate the effectiveness of our training approach, we apply our trained models to various classical image restoration problems. Numerical experiments show that our trained models clearly outperform existing analysis operator learning approaches and are on par with state-of-the-art image denoising algorithms. Our approach develops a framework that is intuitive to understand and easy to implement.
The ability of having a sparse representation for a certain class of signals has many applications in data analysis, image processing, and other research fields. Among sparse representations, the cosparse analysis model has recently gained increasing interest. Many signals exhibit a multidimensional structure, e.g. images or three-dimensional MRI scans. Most data analysis and learning algorithms use vectorized signals and thereby do not account for this underlying structure. The drawback of not taking the inherent structure into account is a dramatic increase in computational cost. We propose an algorithm for learning a cosparse Analysis Operator that adheres to the preexisting structure of the data, and thus allows for a very efficient implementation. This is achieved by enforcing a separable structure on the learned operator. Our learning algorithm is able to deal with multidimensional data of arbitrary order. We evaluate our method on volumetric data at the example of three-dimensional MRI scans.
The paper addresses the image fusion problem, where multiple images captured with different focus distances are to be combined into a higher quality all-in-focus image. Most current approaches for image fusion strongly rely on the unrealistic noise-free assumption used during the image acquisition, and then yield limited robustness in fusion processing. In our approach, we formulate the multi-focus image fusion problem in terms of an analysis sparse model, and simultaneously perform the restoration and fusion of multi-focus images. Based on this model, we propose an analysis operator learning, and define a novel fusion function to generate an all-in-focus image. Experimental evaluations confirm the effectiveness of the proposed fusion approach both visually and quantitatively, and show that our approach outperforms state-of-the-art fusion methods.