Intrinsic Coordinate Research Articles

IntrinsicNGP: Intrinsic Coordinate Based Hash Encoding for Human NeRF.

Recently, many works have been proposed to use the neural radiance field for novel view synthesis of human performers. However, most of these methods require hours of training, making them difficult for practical use. To address this challenging problem, we propose IntrinsicNGP, which can be trained from scratch and achieve high-fidelity results in a few minutes with videos of a human performer. To achieve this goal, we introduce a continuous and optimizable intrinsic coordinate instead of the original explicit euclidean coordinate in the hash encoding module of InstantNGP. With this novel intrinsic coordinate, IntrinsicNGP can aggregate interframe information for dynamic objects using proxy geometry shapes. Moreover, the results trained with the given rough geometry shapes can be further refined with an optimizable offset field based on the intrinsic coordinate. Extensive experimental results on several datasets demonstrate the effectiveness and efficiency of IntrinsicNGP. We also illustrate the ability of our approach to edit the shape of reconstructed objects.

Steric Effect and Intrinsic Electrophilicity and Nucleophilicity from Conceptual Density Functional Theory and Information-Theoretic Approach as Quantitative Probes of Chemical Reactions.

Appreciating reactivity in terms of physicochemical effects for chemical processes is one of the most important undertakings in chemistry. While transition state (TS) theory provides the framework enabling the reliable calculation of the barrier height for a given elementary step, analytical tools are necessary to gain insight into key factors governing the different processes during chemical reactions. In this contribution, we partition the potential energy surface of an elementary step along the intrinsic coordinate into three segments, the so-called Pre-TS, TS, and Post-TS regions, and then determine the most important factors dictating each segment. This analysis is based on the use of both reactivity descriptors from conceptual density functional theory and concepts from the information-theoretic approach in density functional theory. We found that in both Pre-TS and Post-TS regions, steric effects are the dominant factors, whereas in the TS region, it is the intrinsic electrophilic and nucleophilic propensity of the transition state structure that governs the reactivity. The wide applicability of our approach is shown by a validation for a total of 37 organic and inorganic reactions. Different partition approaches from density functional theory, energy decomposition analysis and wavefunction-based resonance theory are employed to support our main conclusions.

Reduced‐Order Modeling for Linearized Representations of Microphysical Process Rates

AbstractRepresenting cloud microphysical processes in large scale atmospheric models is challenging because many processes depend on the details of the droplet size distribution (DSD, the spectrum of droplets with different sizes in a cloud). While full or partial statistical moments of droplet size distributions are the typical variables used in bulk models, prognostic moments are limited in their ability to represent microphysical processes across the range of conditions experienced in the atmosphere. Microphysical parameterizations employing prognostic moments are known to suffer from structural uncertainty in their representations of inherently higher dimensional cloud processes, which limit model fidelity and lead to forecasting errors. Here we investigate how data‐driven reduced‐order modeling can be used to learn predictors for microphysical process rates in bulk microphysics schemes in an unsupervised manner from higher dimensional bin distributions. Using simulations characteristic of marine stratiform clouds, we simultaneously learn lower dimensional representations of droplet size distributions and predict the evolution of the microphysical state of the system. Droplet collision‐coalescence, the main process for generating warm rain, is estimated to have an intrinsic dimension of three. This intrinsic dimension provides a lower limit on the number of degrees of freedom needed to accurately represent collision‐coalescence in models. We demonstrate how deep learning based reduced‐order modeling can be used to discover intrinsic coordinates describing the microphysical state of the system, where process rates such as collision‐coalescence are globally linearized. These implicitly learned representations of the DSD retain more information about the DSD than typical moment‐based representations.

Deep kinematic inference affords efficient and scalable control of bodily movements.

Performing goal-directed movements requires mapping goals from extrinsic (workspace-relative) to intrinsic (body-relative) coordinates and then to motor signals. Mainstream approaches based on optimal control realize the mappings by minimizing cost functions, which is computationally demanding. Instead, active inference uses generative models to produce sensory predictions, which allows a cheaper inversion to the motor signals. However, devising generative models to control complex kinematic chains like the human body is challenging. We introduce an active inference architecture that affords a simple but effective mapping from extrinsic to intrinsic coordinates via inference and easily scales up to drive complex kinematic chains. Rich goals can be specified in both intrinsic and extrinsic coordinates using attractive or repulsive forces. The proposed model reproduces sophisticated bodily movements and paves the way for computationally efficient and biologically plausible control of actuated systems.

Large Deflection and Post-Buckling Analysis of Cantilever Nanorods Including Effects of Couple and Surface Stresses by Intrinsic Coordinate Finite Elements

This paper presents a novel approach to the analysis of large deflection and post-buckling behavior of cantilever nanorods under different load conditions. The proposed approach utilizes a variational method, incorporating the Gurtin–Murdoch surface elasticity theory and the consistent couple stress theory to account for size-dependency effects. By accounting for the strain energy contributions of the bulk material, surface layer, and load conditions, this approach expresses the behavior of the nanorods in terms of their intrinsic coordinates. A finite element method was used to solve this numerical problem, generating a system of non-linear equations that were iteratively solved using the Newton–Raphson method. The results obtained from the finite element method were confirmed by those from the shooting method. Also, it highlights the effectiveness of the variational model in predicting the large deflection and post-buckling behavior of cantilever nanorods while considering both surface stress and couple stress effects. Furthermore, the study investigated the influence of surface stress and couple stress on the response of the nanorods to point and uniform loads. The results showed that incorporating both effects enhanced the stiffness of the nanorods compared to scenarios in which these effects were neglected. In addition, the couple stress effect was found to have a greater influence on the stiffening of nanorods for the same value of unitless parameters compared to surface stress effects. These findings offer valuable insights into the large deflection and post-buckling behavior of cantilever nanorods and highlight the importance of surface stress and couple stress effects. The model proposed in this study has potential applications in advanced technological device designs by providing a comprehensive understanding of the mechanical response of nanorods, allowing for more accurate predictions and enhanced device performance.

Decoding reach-to-grasp from EEG using classifiers trained with data from the contralateral limb.

Fundamental to human movement is the ability to interact with objects in our environment. How one reaches an object depends on the object's shape and intended interaction afforded by the object, e.g., grasp and transport. Extensive research has revealed that the motor intention of reach-to-grasp can be decoded from cortical activities using EEG signals. The goal of the present study is to determine the extent to which information encoded in the EEG signals is shared between two limbs to enable cross-hand decoding. We performed an experiment in which human subjects (n = 10) were tasked to interact with a novel object with multiple affordances using either right or left hands. The object had two vertical handles attached to a horizontal base. A visual cue instructs what action (lift or touch) and whether the left or right handle should be used for each trial. EEG was recorded and processed from bilateral frontal-central-parietal regions (30 channels). We trained LDA classifiers using data from trials performed by one limb and tested the classification accuracy using data from trials performed by the contralateral limb. We found that the type of hand-object interaction can be decoded with approximately 59 and 69% peak accuracy in the planning and execution stages, respectively. Interestingly, the decoding accuracy of the reaching directions was dependent on how EEG channels in the testing dataset were spatially mirrored, and whether directions were labeled in the extrinsic (object-centered) or intrinsic (body-centered) coordinates.

Applications of intrinsic coordinates for simulation of thin shocks in relaxing media

The augmented Burgers equation that describes nonlinear propagation in a relaxing fluid can be expressed using intrinsic coordinates [Hammerton and Crighton, JFM 252, 585 (1993)]. A multivalued pressure waveform in physical coordinates can be represented as single-valued with intrinsic coordinates. Shocks can then be inserted into the multivalued pressure waveform using the equal area rule from weak shock theory to obtain a single-valued waveform solution, thus avoiding the high computational cost associated with discretizing thin shocks with conventional algorithms. By first solving the evolution equation in intrinsic coordinates, and then using the solution as the input to a conventional algorithm based on the augmented Burgers equation, a two-stage approach can be developed that achieves both high accuracy and significantly reduced computational cost when compared to solving the Burgers equation with conventional numerical algorithms alone. Approaches based on intrinsic coordinates are applied to realistic problems in relaxing media, which include air and seawater, to allow for comparisons of computational efficiency and accuracy. [WAW is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

BPM: Blended Piecewise Möbius Maps

AbstractWe propose a novel Möbius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a continuous map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi‐conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed‐form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi‐conformal errors.

Differential processing of intrinsic vs. extrinsic coordinates in wrist movement: connectivity and chronometry perspectives.

This study explores brain-network differences between the intrinsic and extrinsic motor coordinate frames. A connectivity model showing the coordinate frames difference was obtained using brain fMRI data of right wrist isometric flexions and extensions movements, performed in two forearm postures. The connectivity model was calculated by machine-learning-based neural representation and effective functional connectivity using psychophysiological interaction and dynamic causal modeling analyses. The model indicated the network difference wherein the inferior parietal lobule receives extrinsic information from the rostral lingual gyrus through the superior parietal lobule and transmits intrinsic information to the Handknob, whereas extrinsic information is transmitted to the Handknob directly from the rostral lingual gyrus. A behavioral experiment provided further evidence on the difference between motor coordinate frames showing onset timing delay of muscle activity of intrinsic coordinate-directed wrist movement compared to extrinsic one. These results suggest that, if the movement is externally directed, intrinsic coordinate system information is bypassed to reach the primary motor area.

Bilateral transfer of motor performance as a function of motor imagery training: a systematic review and meta-analysis.

The objective of this review was to evaluate the efficacy of mental imagery training (MIT) in promoting bilateral transfer (BT) of motor performance for healthy subjects. We searched 6 online-databases (Jul-Dec 2022) using terms: "mental practice," "motor imagery training," "motor imagery practice," "mental training," "movement imagery," "cognitive training," "bilateral transfer," "interlimb transfer," "cross education," "motor learning," "strength," "force" and "motor performance." We selected randomized-controlled studies that examined the effect of MIT on BT. Two reviewers independently determined if each study met the inclusion criteria for the review. Disagreements were resolved through discussion and, if necessary, by a third reviewer. A total of 9 articles out of 728 initially identified studies were chosen for the meta-analysis. The meta-analysis included 14 studies for the comparison between MIT and no-exercise control (CTR) and 15 studies for the comparison between MIT and physical training (PT). MIT showed significant benefit in inducing BT compared to CTR (ES = 0.78, 95% CI = 0.57-0.98). The effect of MIT on BT was similar to that of PT (ES = -0.02, 95% CI = -0.15-0.17). Subgroup analyses showed that internal MIT (IMIT) was more effective (ES = 2.17, 95% CI = 1.57-2.76) than external MIT (EMIT) (ES = 0.95, 95% CI = 0.74-1.17), and mixed-task (ES = 1.68, 95% CI = 1.26-2.11) was more effective than mirror-task (ES = 0.46, 95% CI = 0.14-0.78) and normal-task (ES = 0.56, 95% CI = 0.23-0.90). No significant difference was found between transfer from dominant limb (DL) to non-dominant limb (NDL) (ES = 0.67, 95% CI = 0.37-0.97) and NDL to DL (ES = 0.87, 95% CI = 0.59-1.15). This review concludes that MIT can serve as a valuable alternative or supplement to PT in facilitating BT effects. Notably, IMIT is preferable to EMIT, and interventions incorporating tasks that have access to both intrinsic and extrinsic coordinates (mixed-task) are preferred over those that involve only one of the two coordinates (mirror-task or normal-task). These findings have implications for rehabilitation of patients such as stroke survivors.

A study on data-driven identification and representation of nonlinear dynamical systems with a physics-integrated deep learning approach: Koopman operators and nonlinear normal modes

Identifying the intrinsic coordinates or modes of dynamical systems is essential to understand, analyze, and characterize the underlying complex dynamical behaviors. This is a critical challenge for nonlinear dynamical systems because the modal transformation, which is universal for linear systems, no longer applies. Koopman operators and nonlinear normal modes (NNMs) are two main frameworks aiming to provide a general representation of nonlinear dynamics. In this study, we investigate their abilities on representing nonlinear dynamical systems. Addressing the challenge that the closed-form models or equations of dynamical systems are generally unknown realistically, we first present a physics-integrated deep learning-based data-driven framework to identify the eigenfunction of Koopman operators and the nonlinear modal transformation function of NNMs, respectively. We then evaluate their representation abilities by their reconstruction and predictive accuracy of the nonlinear dynamical response using the identified Koopman modes and NNMs. We conduct numerical experiments on Duffing systems with varying nonlinearity levels and dimensions, and observe that NNMs achieve higher representation accuracy and better computational efficiency than Koopman operators using the same dimension of intrinsic coordinates or modes.

Model order reduction for parameterized electromagnetic problems using matrix decomposition and deep neural networks

A non-intrusive model order reduction (MOR) method for solving parameterized electromagnetic scattering problems is proposed in this paper. A database collecting snapshots of high-fidelity solutions is built by solving the parameterized time-domain Maxwell equations for some values of the material parameters using a fullwave solver based on a high order discontinuous Galerkin time-domain (DGTD) method. To perform a prior dimensionality reduction, a set of reduced basis (RB) functions are extracted from the database via a two-step proper orthogonal decomposition (POD) method. Intrinsic coordinates of the high-fidelity solutions are further compressed through a convolutional autoencoder (CAE) network. Singular value decomposition (SVD) is then used to extract the principal components of the low dimensional coding matrices generated by CAE, and a cubic spline interpolation-based (CSI) approach is employed for approximating the dominating time- and parameter-modes of these matrices. The generation of the reduced basis and the training of the CAE and CSI are accomplished in the offline stage, thus the RB solution for given time/parameter values can be quickly recovered via outputs of the interpolation model and decoder network. In particular, the offline and online stages of the proposed RB method are completely decoupled, which ensures the validity of the method. The performance of the proposed CAE-CSI ROM is illustrated with numerical experiments for scattering of a plane wave by a 2-D dielectric disk and a multi-layer heterogeneous medium.

Modeling of Four-Winged Micro Ornithopters Inspired by Dragonflies

In this paper, we present a full dynamical model of a four-winged micro ornithopter inspired by a dragonfly-type insect. The micro ornithopter is modeled as four articulated rigid body components (wings) connected to the main body via spherical joints. The dynamical model is derived using Lagrangian mechanics with intrinsic global coordinates, without relying on the common assumptions that neglect the wings-body interactions. Furthermore, the aerodynamic forces are modeled under the quasi-steady motion assumption without restricting the flapping frequency to be relatively high. This provides a full and elegant four-winged micro ornithopter model that captures the interaction between the body and the wings while avoiding the complexities and singularities associated with other coordinate representations (e.g., Euler angles). Simulation studies of the inertial effects of the relative motion between the different parts of the multibody system show the importance of considering the forces and torques, resulting from the wings-body interaction, in motion generation of these insects.

The Marcus dimension: identifying the nuclear coordinate for electron transfer from ab initio calculations.

The Marcus model forms the foundation for all modern discussion of electron transfer (ET). In this model, ET results in a change in diabatic potential energy surfaces, separated along an ET nuclear coordinate. This coordinate accounts for all nuclear motion that promotes electron transfer. It is usually assumed to be dominated by a collective asymmetric vibrational motion of the redox sites involved in the ET. However, this coordinate is rarely quantitatively specified. Instead, it remains a nebulous concept, rather than a tool for gaining true insight into the ET pathway. Herein, we describe an ab initio approach for quantifying the ET coordinate and demonstrate it for a series of dinitroradical anions. Using sampling methods at finite temperature combined with density functional theory calculations, we find that the electron transfer can be followed using the energy separation between potential energy surfaces and the extent of electron localization. The precise nuclear motion that leads to electron transfer is then obtained as a linear combination of normal modes. Once the coordinate is identified, we find that evolution along it results in a change in diabatic state and optical excitation energy, as predicted by the Marcus model. Thus, we conclude that a single dimension of the electron transfer described in Marcus-Hush theory can be described as a well-defined nuclear motion. Importantly, our approach allows the separation of the intrinsic electron transfer coordinate from other structural relaxations and environmental influences. Furthermore, the barrier separating the adiabatic minima was found to be sufficiently thin to enable heavy-atom tunneling in the ET process.

Koopman operator for time-dependent reliability analysis

Time-dependent structural reliability analysis of nonlinear dynamical systems is non-trivial; subsequently, scope of most of the structural reliability analysis methods is limited to time-independent reliability analysis only. In this work, we propose a Koopman operator based approach for time-dependent reliability analysis of nonlinear dynamical systems. Since the Koopman representations can transform any nonlinear dynamical system into a linear dynamical system, the time evolution of dynamical systems can be obtained by Koopman operators seamlessly regardless of the nonlinear or chaotic behavior. Despite the fact that the Koopman theory has been in vogue a long time back, identifying intrinsic coordinates is a challenging task; to address this, we propose an end-to-end deep learning architecture that learns the Koopman observables and then use it for time marching the dynamical response. Unlike purely data-driven approaches, the proposed approach is robust even in the presence of uncertainties; this renders the proposed approach suitable for time-dependent reliability analysis. We propose two architectures; one suitable for time-dependent reliability analysis when the system is subjected to random initial condition and the other suitable when the underlying system have uncertainties in system parameters. The proposed approach is robust and generalizes to unseen environment (out-of-distribution prediction). Efficacy of the proposed approached is illustrated using four numerical examples. Results obtained indicate supremacy of the proposed approach as compared to purely data-driven auto-regressive neural network, long short- term memory network, and DeepONet.

Normal form for pseudo-Einstein contact forms and intrinsic CR normal coordinates

We give a normal form for pseudo-Einstein contact forms and apply it to construct intrinsic CR normal coordinates parametrized by the structure group of CR geometry. The proof is based on the construction of parabolic normal coordinates by Jerison and Lee.

Deep multi-modal learning for joint linear representation of nonlinear dynamical systems

Dynamical systems pervasively seen in most real-life applications are complex and behave by following certain evolution rules or dynamical patterns, which are linear, non-linear, or stochastic. The underlying dynamics (or evolution rule) of such complex systems, if found, can be used for understanding the system behavior, and furthermore for system prediction and control. It is common to analyze the system’s dynamics through observations in different modality approaches. For instance, to recognize patient deterioration in acute care, it usually relies on monitoring and analyzing vital signs and other observations, such as blood pressure, heart rate, respiration, and electroencephalography. These observations convey the information describing the same target system, but the dynamics is not able to be directly characterized due to high complexity of individual modality and maybe time-delay interactions among modalities. In this work, we suppose that the state behavior of a dynamical system follows an intrinsic dynamics shared among these modalities. We specifically propose a new deep auto-encoder framework using the Koopman operator theory to derive the joint linear dynamics for a target system in a space spanned by the intrinsic coordinates. The proposed method aims to reconstruct the original system states by learning the information provided among multiple modalities. Furthermore, with the derived intrinsic dynamics, our method is capable of restoring the missing observations within and across modalities, and used for predicting the future states of the system that follows the same evolution rule.

A Convex Relaxation Approach for Learning the Robust Koopman Operator

Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow.

Critical Field Theories with OSp(1|2M) Symmetry

In the paper [L. Fei etal., J. High Energy Phys. 09 (2015) 076JHEPFG1029-847910.1007/JHEP09(2015)076] a cubic field theory of a scalar field σ and two anticommuting scalar fields, θ and θ[over ¯], was formulated. In 6-ε dimensions it has a weakly coupled fixed point with imaginary cubic couplings where the symmetry is enhanced to the supergroup OSp(1|2). This theory may be viewed as a "UV completion" in 2<d<6 of the nonlinear sigma model with hyperbolic target space H^{0|2} described by a pair of intrinsic anticommuting coordinates. It also describes the q→0 limit of the critical q-state Potts model, which is equivalent to the statistical mechanics of spanning forests on a graph. In this Letter we generalize these results to a class of OSp(1|2M) symmetric field theories whose upper critical dimensions are d_{c}(M)=2[(2M+1)/(2M-1)]. They contain 2M anticommuting scalar fields, θ^{i}, θ[over ¯]^{i}, and one commuting one, with interaction g(σ^{2}+2θ^{i}θ[over ¯]^{i})^{(2M+1)/2}. Ind_{c}(M)-ε dimensions, we find a weakly coupled IR fixed point at an imaginary value of g. We propose that these critical theories are the UV completions of the sigma models with fermionic hyperbolic target spaces H^{0|2M}. Of particular interest is the quintic field theory with OSp(1|4) symmetry, whose upper critical dimension is 10/3. Using this theory, we make a prediction for the critical behavior of the OSp(1|4) lattice system in three dimensions.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.