DNA Microscopy: Optics-free Spatio-genetic Imaging by a Stand-Alone Chemical Reaction.

Analyzing the spatial organization of molecules in cells and tissues is a cornerstone of biological research and clinical practice. However, despite enormous progress in molecular profiling of cellular constituents, spatially mapping them remains a disjointed and specialized machinery-intensive process, relying on either light microscopy or direct physical registration. Here, we demonstrate DNA microscopy, a distinct imaging modality for scalable, optics-free mapping of relative biomolecule positions. In DNA microscopy of transcripts, transcript molecules are tagged in situ with randomized nucleotides, labeling each molecule uniquely. A second in situ reaction then amplifies the tagged molecules, concatenates the resulting copies, and adds new randomized nucleotides to uniquely label each concatenation event. An algorithm decodes molecular proximities from these concatenated sequences and infers physical images of the original transcripts at cellular resolution with precise sequence information. Because its imaging power derives entirely from diffusive molecular dynamics, DNA microscopy constitutes a chemically encoded microscopy system.

Stable diverse food webs become more common when interactions are more biologically constrained.

Ecologists have long sought to understand how diversity and structure mediate the stability of whole ecosystems. For high-diversity food webs, the interactions between species are typically represented using matrices with randomly chosen interaction strengths. Unfortunately, this procedure tends to produce ecological systems with no underlying equilibrium solution, and so ecological inferences from this approach may be biased by nonbiological outcomes. Using recent computationally efficient methodological advances from metabolic networks, we employ for the first time an inverse approach to diversity-stability research. We compare classical random interaction matrices of realistic food web topology (hereafter the classical model) to feasible, biologically constrained, webs produced using the inverse approach. We show that an energetically constrained feasible model yields a far higher proportion of stable high-diversity webs than the classical random matrix approach. When we examine the energetically constrained interaction strength distributions of these matrix models, we find that although these diverse webs have consistent negative self-regulation, they do not require strong self-regulation to persist. These energetically constrained diverse webs instead show an increasing preponderance of weak interactions that are known to increase local stability. Further examination shows that some of these weak interactions naturally appear to arise in the model food webs from a constraint-generated realistic generalist-specialist trade-off, whereby generalist predators have weaker interactions than more specialized species. Additionally, the inverse technique we present here has enormous promise for understanding the role of the biological structure behind stable high-diversity webs and for linking empirical data to the theory.

Elliptic PDE learning is provably data-efficient.

Partial differential equations (PDE) learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of three-dimensional uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.

Revisiting [Formula: see text]-wavelet compressed-sensing MRI in the era of deep learning.

Following their success in numerous imaging and computer vision applications, deep-learning (DL) techniques have emerged as one of the most prominent strategies for accelerated MRI reconstruction. These methods have been shown to outperform conventional regularized methods based on compressed sensing (CS). However, in most comparisons, CS is implemented with two or three hand-tuned parameters, while DL methods enjoy a plethora of advanced data science tools. In this work, we revisit [Formula: see text]-wavelet CS reconstruction using these modern tools. Using ideas such as algorithm unrolling and advanced optimization methods over large databases that DL algorithms utilize, along with conventional insights from wavelet representations and CS theory, we show that [Formula: see text]-wavelet CS can be fine-tuned to a level close to DL reconstruction for accelerated MRI. The optimized [Formula: see text]-wavelet CS method uses only 128 parameters compared to >500,000 for DL, employs a convex reconstruction at inference time, and performs within <1% of a DL approach that has been used in multiple studies in terms of quantitative quality metrics.

The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem.

Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities; however, there does not exist any algorithm, even randomized, that can train (or compute) such a NN. For any positive integers K>2 and L, there are cases where simultaneously 1) no randomized training algorithm can compute a NN correct to K digits with probability greater than 1/2; 2) there exists a deterministic training algorithm that computes a NN with K ­1 correct digits, but any such (even randomized) algorithm needs arbitrarily many training data; and 3) there exists a deterministic training algorithm that computes a NN with K ­2 correct digits using no more than L training samples. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. We introduce fast iterative restarted networks (FIRENETs), which we both prove and numerically verify are stable. Moreover, we prove that only O(|log (ϵ)|) layers are needed for an ϵ-accurate solution to the inverse problem.

Self-supervised learning for improved calibrationless radial MRI with NLINV-Net.

PURPOSE: To develop a neural network architecture for improved calibrationless reconstruction of radial data when no ground truth is available for training. METHODS: NLINV-Net is a model-based neural network architecture that directly estimates images and coil sensitivities from (radial) k-space data via nonlinear inversion (NLINV). Combined with a training strategy using self-supervision via data undersampling (SSDU), it can be used for imaging problems where no ground truth reconstructions are available. We validated the method for (1) real-time cardiac imaging and (2) single-shot subspace-based quantitative T1 mapping. Furthermore, region-optimized virtual (ROVir) coils were used to suppress artifacts stemming from outside the field of view and to focus the k-space-based SSDU loss on the region of interest. NLINV-Net-based reconstructions were compared with conventional NLINV and PI-CS (parallel imaging + compressed sensing) reconstruction and the effect of the region-optimized virtual coils and the type of training loss was evaluated qualitatively. RESULTS: NLINV-Net-based reconstructions contain significantly less noise than the NLINV-based counterpart. ROVir coils effectively suppress streakings which are not suppressed by the neural networks while the ROVir-based focused loss leads to visually sharper time series for the movement of the myocardial wall in cardiac real-time imaging. For quantitative imaging, T1-maps reconstructed using NLINV-Net show similar quality as PI-CS reconstructions, but NLINV-Net does not require slice-specific tuning of the regularization parameter. CONCLUSION: NLINV-Net is a versatile tool for calibrationless imaging which can be used in challenging imaging scenarios where a ground truth is not available.

SPICER: Self-supervised learning for MRI with automatic coil sensitivity estimation and reconstruction.

PURPOSE: To introduce a novel deep model-based architecture (DMBA), SPICER, that uses pairs of noisy and undersampled k-space measurements of the same object to jointly train a model for MRI reconstruction and automatic coil sensitivity estimation. METHODS: SPICER consists of two modules to simultaneously reconstructs accurate MR images and estimates high-quality coil sensitivity maps (CSMs). The first module, CSM estimation module, uses a convolutional neural network (CNN) to estimate CSMs from the raw measurements. The second module, DMBA-based MRI reconstruction module, forms reconstructed images from the input measurements and the estimated CSMs using both the physical measurement model and learned CNN prior. With the benefit of our self-supervised learning strategy, SPICER can be efficiently trained without any fully sampled reference data. RESULTS: We validate SPICER on both open-access datasets and experimentally collected data, showing that it can achieve state-of-the-art performance in highly accelerated data acquisition settings (up to 10 × $$ 10\times $$ ). Our results also highlight the importance of different modules of SPICER-including the DMBA, the CSM estimation, and the SPICER training loss-on the final performance of the method. Moreover, SPICER can estimate better CSMs than pre-estimation methods especially when the ACS data is limited. CONCLUSION: Despite being trained on noisy undersampled data, SPICER can reconstruct high-quality images and CSMs in highly undersampled settings, which outperforms other self-supervised learning methods and matches the performance of the well-known E2E-VarNet trained on fully sampled ground-truth data.

Learning hidden elasticity with deep neural networks.

Elastography is an imaging technique to reconstruct elasticity distributions of heterogeneous objects. Since cancerous tissues are stiffer than healthy ones, for decades, elastography has been applied to medical imaging for noninvasive cancer diagnosis. Although the conventional strain-based elastography has been deployed on ultrasound diagnostic-imaging devices, the results are prone to inaccuracies. Model-based elastography, which reconstructs elasticity distributions by solving an inverse problem in elasticity, may provide more accurate results but is often unreliable in practice due to the ill-posed nature of the inverse problem. We introduce ElastNet, a de novo elastography method combining the theory of elasticity with a deep-learning approach. With prior knowledge from the laws of physics, ElastNet can escape the performance ceiling imposed by labeled data. ElastNet uses backpropagation to learn the hidden elasticity of objects, resulting in rapid and accurate predictions. We show that ElastNet is robust when dealing with noisy or missing measurements. Moreover, it can learn probable elasticity distributions for areas even without measurements and generate elasticity images of arbitrary resolution. When both strain and elasticity distributions are given, the hidden physics in elasticity-the conditions for equilibrium-can be learned by ElastNet.

Nonparametric coalescent inference of mutation spectrum history and demography.

As populations boom and bust, the accumulation of genetic diversity is modulated, encoding histories of living populations in present-day variation. Many methods exist to decode these histories, and all must make strong model assumptions. It is typical to assume that mutations accumulate uniformly across the genome at a constant rate that does not vary between closely related populations. However, recent work shows that mutational processes in human and great ape populations vary across genomic regions and evolve over time. This perturbs the mutation spectrum (relative mutation rates in different local nucleotide contexts). Here, we develop theoretical tools in the framework of Kingman's coalescent to accommodate mutation spectrum dynamics. We present mutation spectrum history inference (mushi), a method to perform nonparametric inference of demographic and mutation spectrum histories from allele frequency data. We use mushi to reconstruct trajectories of effective population size and mutation spectrum divergence between human populations, identify mutation signatures and their dynamics in different human populations, and calibrate the timing of a previously reported mutational pulse in the ancestors of Europeans. We show that mutation spectrum histories can be placed in a well-studied theoretical setting and rigorously inferred from genomic variation data, like other features of evolutionary history.

Simple MATLAB and Python scripts for multi-exponential analysis.

Multi-exponential decay is prevalent in magnetic resonance spectroscopy, relaxation, and imaging. This paper describes simple MATLAB and Python functions and scripts for regularized multi-exponential analysis methods for 1D and 2D data and example test problems and experiments. Regularized least-squares solutions provide production-quality outputs with robust stopping rules in ~5 and ~20 lines of code for 1D and 2D inversions, respectively. The software provides an open-architecture simple solution for transforming exponential decay data to the distribution of their decay lifetimes. Examples from magnetic resonance relaxation of a complex fluid, a Danish North Sea crude oil, and fluid mixtures in porous materials-brine/crude oil mixture in North Sea reservoir chalk-are presented. Developed codes may be incorporated in other software or directly used by other researchers, in magnetic resonance relaxation, diffusion, and imaging or other physical phenomena that require multi-exponential analysis.

Magnetic Sensor Array for Electric Arc Reconstruction in Circuit Breakers.

Noninvasive imaging of circuit breakers under short-circuit testing is addressed by recording the magnetic field produced over an array of external sensors and by solving an inverse problem to identify the causing current distribution. The temporal and spatial resolution of the sensing chain are studied and implemented in a physical set-up. A wire model is adopted to describe electrical current distribution. Additionally, the simpler, more direct approach to evaluating the passage of electric current in front of sensors is proposed. The dynamics of suitable approximating models of the electric arc that forms across contacts is obtained and agrees with multi-physical simulations and with experimental time histories of current and voltage. The two methods are flexible and allow the analysis of different types of circuit breakers.

Robust Reconstruction of the Void Fraction from Noisy Magnetic Flux Density Using Invertible Neural Networks.

Electrolysis stands as a pivotal method for environmentally sustainable hydrogen production. However, the formation of gas bubbles during the electrolysis process poses significant challenges by impeding the electrochemical reactions, diminishing cell efficiency, and dramatically increasing energy consumption. Furthermore, the inherent difficulty in detecting these bubbles arises from the non-transparency of the wall of electrolysis cells. Additionally, these gas bubbles induce alterations in the conductivity of the electrolyte, leading to corresponding fluctuations in the magnetic flux density outside of the electrolysis cell, which can be measured by externally placed magnetic sensors. By solving the inverse problem of the Biot-Savart Law, we can estimate the conductivity distribution as well as the void fraction within the cell. In this work, we study different approaches to solve the inverse problem including Invertible Neural Networks (INNs) and Tikhonov regularization. Our experiments demonstrate that INNs are much more robust to solving the inverse problem than Tikhonov regularization when the level of noise in the magnetic flux density measurements is not known or changes over space and time.

Iteratively Refined Image Reconstruction with Learned Attentive Regularizers.

We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.

Deep, deep learning with BART.

PURPOSE: To develop a deep-learning-based image reconstruction framework for reproducible research in MRI. METHODS: The BART toolbox offers a rich set of implementations of calibration and reconstruction algorithms for parallel imaging and compressed sensing. In this work, BART was extended by a nonlinear operator framework that provides automatic differentiation to allow computation of gradients. Existing MRI-specific operators of BART, such as the nonuniform fast Fourier transform, are directly integrated into this framework and are complemented by common building blocks used in neural networks. To evaluate the use of the framework for advanced deep-learning-based reconstruction, two state-of-the-art unrolled reconstruction networks, namely the Variational Network and MoDL, were implemented. RESULTS: State-of-the-art deep image-reconstruction networks can be constructed and trained using BART's gradient-based optimization algorithms. The BART implementation achieves a similar performance in terms of training time and reconstruction quality compared to the original implementations based on TensorFlow. CONCLUSION: By integrating nonlinear operators and neural networks into BART, we provide a general framework for deep-learning-based reconstruction in MRI.

Bayesian MRI reconstruction with joint uncertainty estimation using diffusion models.

PURPOSE: We introduce a framework that enables efficient sampling from learned probability distributions for MRI reconstruction. METHOD: Samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method, different from conventional deep learning-based MRI reconstruction techniques. In addition to the maximum a posteriori estimate for the image, which can be obtained by maximizing the log-likelihood indirectly or directly, the minimum mean square error estimate and uncertainty maps can also be computed from those drawn samples. The data-driven Markov chains are constructed with the score-based generative model learned from a given image database and are independent of the forward operator that is used to model the k-space measurement. RESULTS: We numerically investigate the framework from these perspectives: (1) the interpretation of the uncertainty of the image reconstructed from undersampled k-space; (2) the effect of the number of noise scales used to train the generative models; (3) using a burn-in phase in MCMC sampling to reduce computation; (4) the comparison to conventional ℓ 1 $$ {\ell}_1 $$ -wavelet regularized reconstruction; (5) the transferability of learned information; and (6) the comparison to fastMRI challenge. CONCLUSION: A framework is described that connects the diffusion process and advanced generative models with Markov chains. We demonstrate its flexibility in terms of contrasts and sampling patterns using advanced generative priors and the benefits of also quantifying the uncertainty for every pixel.

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data.

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs)-and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of Escherichia coli bacterial motility. The fine scale, chemomechanical, hybrid (continuum-Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained "Keller-Segel class" chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and "hardwired" in the regression process. More importantly, we discuss data-driven corrections (both additive and functional), to analytically known, approximate closures.

On instabilities of deep learning in image reconstruction and the potential costs of AI.

Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field. In this paper, we demonstrate a crucial phenomenon: Deep learning typically yields unstable methods for image reconstruction. The instabilities usually occur in several forms: 1) Certain tiny, almost undetectable perturbations, both in the image and sampling domain, may result in severe artefacts in the reconstruction; 2) a small structural change, for example, a tumor, may not be captured in the reconstructed image; and 3) (a counterintuitive type of instability) more samples may yield poorer performance. Our stability test with algorithms and easy-to-use software detects the instability phenomena. The test is aimed at researchers, to test their networks for instabilities, and for government agencies, such as the Food and Drug Administration (FDA), to secure safe use of deep learning methods.

Developments of inverse analysis by Kalman filters and Bayesian methods applied to geotechnical engineering.

The present paper reviews recent activities on inverse analysis strategies in geotechnical engineering using Kalman filters, nonlinear Kalman filters, and Markov chain Monte Carlo (MCMC)/Hamiltonian Monte Carlo (HMC) methods. Nonlinear Kalman filters with finite element method (FEM) broaden the choices of unknowns to be determined for not only parameters but also initial and/or boundary conditions, and the use of the posterior probability of the state variables can be widely applied to, for example, the decision making for design changes. The relevance of the unknowns and the observed values and the selection of the best sensor locations are some of the considerations made while using the Kalman filter FEM. This paper demonstrates several real-world geotechnical applications of the nonlinear Kalman filter and the MCMC with FEM. Future studies should focus on the following areas: attaining excellent performance for long-term forecasts using short-term observation and developing a viable method for selecting equations that describe physical phenomena and constitutive models.

On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition Numbers.

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree of ambiguity that may exist about its solution. Traditional measures of ill-posedness, such as the condition number of a matrix, provide characterizations that are global in nature. While such characterizations can be powerful, they can also fail to provide full insight into situations where certain entries of the solution vector are more or less ambiguous than others. In this work, we derive novel theoretical lower- and upper-bounds that apply to individual entries of the solution vector, and are valid for all potential solution vectors that are nearly data-consistent. These bounds are agnostic to the noise statistics and the specific method used to solve the inverse problem, and are also shown to be tight. In addition, our results also lead us to introduce an entrywise version of the traditional condition number, which provides a substantially more nuanced characterization of scenarios where certain elements of the solution vector are less sensitive to perturbations than others. Our results are illustrated in an application to magnetic resonance imaging reconstruction, and we include discussions of practical computation methods for large-scale inverse problems, connections between our new theory and the traditional Cramér-Rao bound under statistical modeling assumptions, and potential extensions to cases involving constraints beyond just data-consistency.

A Probabilistic Digital Twin for Leak Localization in Water Distribution Networks Using Generative Deep Learning.

Localizing leakages in large water distribution systems is an important and ever-present problem. Due to the complexity originating from water pipeline networks, too few sensors, and noisy measurements, this is a highly challenging problem to solve. In this work, we present a methodology based on generative deep learning and Bayesian inference for leak localization with uncertainty quantification. A generative model, utilizing deep neural networks, serves as a probabilistic surrogate model that replaces the full equations, while at the same time also incorporating the uncertainty inherent in such models. By embedding this surrogate model into a Bayesian inference scheme, leaks are located by combining sensor observations with a model output approximating the true posterior distribution for possible leak locations. We show that our methodology enables producing fast, accurate, and trustworthy results. It showed a convincing performance on three problems with increasing complexity. For a simple test case, the Hanoi network, the average topological distance (ATD) between the predicted and true leak location ranged from 0.3 to 3 with a varying number of sensors and level of measurement noise. For two more complex test cases, the ATD ranged from 0.75 to 4 and from 1.5 to 10, respectively. Furthermore, accuracies upwards of 83%, 72%, and 42% were achieved for the three test cases, respectively. The computation times ranged from 0.1 to 13 s, depending on the size of the neural network employed. This work serves as an example of a digital twin for a sophisticated application of advanced mathematical and deep learning techniques in the area of leak detection.