Inspired by the necessity of morphological adaptation in animals, a growing body of work has attempted to expand robot training to encompass physical aspects of a robot's design. However, reinforcement learning methods capable of optimizing the 3D morphology of a robot have been restricted to reorienting or resizing the limbs of a predetermined and static topological genus. Here we show policy gradients for designing freeform robots with arbitrary external and internal structure. This is achieved through actions that deposit or remove bundles of atomic building blocks to form higher-level nonparametric macrostructures such as appendages, organs and cavities. Although results are provided for open loop control only, we discuss how this method could be adapted for closed loop control and sim2real transfer to physical machines in future.
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Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.
We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.
A Variable Parameter (VP) analysis aims to give precise time complexity expressions of algorithms with exponents appearing solely in terms of variable parameters. A variable parameter is the number of objects with specific properties. Here we describe two VP-algorithms, an implicit enumeration and a polynomial-time approximation scheme for a strongly $NP$-hard problem of scheduling $n$ independent jobs with release and due times on one machine to minimize the maximum job completion time $C_{\max}$. Thus variable parameters are amounts of some specially defined types of jobs. A partial solution without these jobs is constructed in a low degree polynomial time, and an exponential time (in the number of variable parameters) procedure is carried out to augment this solution to a complete optimal solution. We also give alternative time complexity expressions, where the exponential dependence is solely on some job parameters. Applying the fixed parameter analysis to these estimations, unexpectedly, we obtain a polynomial-time dependence. Both, intuitive probabilistic estimations and our extensive experimental study support our conjecture that the total number of the variable parameters is far less than $n$ and its ratio to $n$ asymptotically converges to 0.
We develop a theory characterizing the fundamental capability of deep neural networks to learn, from evolution traces, the logical rules governing the behavior of cellular automata (CA). This is accomplished by first establishing a novel connection between CA and Lukasiewicz propositional logic. While binary CA have been known for decades to essentially perform operations in Boolean logic, no such relationship exists for general CA. We demonstrate that many-valued (MV) logic, specifically Lukasiewicz propositional logic, constitutes a suitable language for characterizing general CA as logical machines. This is done by interpolating CA transition functions to continuous piecewise linear functions, which, by virtue of the McNaughton theorem, yield formulae in MV logic characterizing the CA. Recognizing that deep rectified linear unit (ReLU) networks realize continuous piecewise linear functions, it follows that these formulae are naturally extracted from CA evolution traces by deep ReLU networks. A corresponding algorithm together with a software implementation is provided. Finally, we show that the dynamical behavior of CA can be realized by recurrent neural networks.
A tremendous range of design tasks in materials, physics, and biology can be formulated as finding the optimum of an objective function depending on many parameters without knowing its closed-form expression or the derivative. Traditional derivative-free optimization techniques often rely on strong assumptions about objective functions, thereby failing at optimizing non-convex systems beyond 100 dimensions. Here, we present a tree search method for derivative-free optimization that enables accelerated optimal design of high-dimensional complex systems. Specifically, we introduce stochastic tree expansion, dynamic upper confidence bound, and short-range backpropagation mechanism to evade local optimum, iteratively approximating the global optimum using machine learning models. This development effectively confronts the dimensionally challenging problems, achieving convergence to global optima across various benchmark functions up to 2,000 dimensions, surpassing the existing methods by 10- to 20-fold. Our method demonstrates wide applicability to a wide range of real-world complex systems spanning materials, physics, and biology, considerably outperforming state-of-the-art algorithms. This enables efficient autonomous knowledge discovery and facilitates self-driving virtual laboratories. Although we focus on problems within the realm of natural science, the advancements in optimization techniques achieved herein are applicable to a broader spectrum of challenges across all quantitative disciplines.
Joint relation modeling is a curial component in human motion prediction. Most existing methods rely on skeletal-based graphs to build the joint relations, where local interactive relations between joint pairs are well learned. However, the motion coordination, a global joint relation reflecting the simultaneous cooperation of all joints, is usually weakened because it is learned from part to whole progressively and asynchronously. Thus, the final predicted motions usually appear unrealistic. To tackle this issue, we learn a medium, called coordination attractor (CA), from the spatiotemporal features of motion to characterize the global motion features, which is subsequently used to build new relative joint relations. Through the CA, all joints are related simultaneously, and thus the motion coordination of all joints can be better learned. Based on this, we further propose a novel joint relation modeling module, Comprehensive Joint Relation Extractor (CJRE), to combine this motion coordination with the local interactions between joint pairs in a unified manner. Additionally, we also present a Multi-timescale Dynamics Extractor (MTDE) to extract enriched dynamics from the raw position information for effective prediction. Extensive experiments show that the proposed framework outperforms state-of-the-art methods in both short- and long-term predictions on H3.6M, CMU-Mocap, and 3DPW.
This work presents a comparative review and classification between some well-known thermodynamically consistent models of hydrogel behavior in a large deformation setting, specifically focusing on solvent absorption/desorption and its impact on mechanical deformation and network swelling. The proposed discussion addresses formulation aspects, general mathematical classification of the governing equations, and numerical implementation issues based on the finite element method. The theories are presented in a unified framework demonstrating that, despite not being evident in some cases, all of them follow equivalent thermodynamic arguments. A detailed numerical analysis is carried out where Taylor-Hood elements are employed in the spatial discretization to satisfy the inf-sup condition and to prevent spurious numerical oscillations. The resulting discrete problems are solved using the FEniCS platform through consistent variational formulations, employing both monolithic and staggered approaches. We conduct benchmark tests on various hydrogel structures, demonstrating that major differences arise from the chosen volumetric response of the hydrogel. The significance of this choice is frequently underestimated in the state-of-the-art literature but has been shown to have substantial implications on the resulting hydrogel behavior.
In this pioneering study, we unveiled a groundbreaking approach for actuating rehabilitation robots through the innovative use of magnetic technology as a seamless haptic force generator, offering a leap forward in enhancing user interface and experience, particularly in end-effector-based robots for upper-limb extremity motor rehabilitation. We employed the Extended Kalman Filter to meticulously analyze and formalize the robotic system's nonlinear dynamics, showcasing the potential of this sophisticated algorithm in accurately tracking and compensating for disturbances, thereby ensuring seamless and effective motor training. The proposed planar robotic system embedded with magnetic technology was evaluated with the recruitment of human subjects. We reached a minimum RMS value of 0.2 and a maximum of 2.06 in our estimations, indicating our algorithm's capability for tracking the system behavior. Overall, the results showed significant improvement in smoothness, comfort, and safety during execution and motor training. The proposed novel magnetic actuation and advanced algorithmic control opens new horizons for the development of more efficient and user-friendly rehabilitation technologies.
Quantifying spatial and/or temporal associations in multivariate geolocated data of different types is achievable via spatial random effects in a Bayesian hierarchical model, but severe computational bottlenecks arise when spatial dependence is encoded as a latent Gaussian process (GP) in the increasingly common large scale data settings on which we focus. The scenario worsens in non-Gaussian models because the reduced analytical tractability leads to additional hurdles to computational efficiency. In this article, we introduce Bayesian models of spatially referenced data in which the likelihood or the latent process (or both) are not Gaussian. First, we exploit the advantages of spatial processes built via directed acyclic graphs, in which case the spatial nodes enter the Bayesian hierarchy and lead to posterior sampling via routine Markov chain Monte Carlo (MCMC) methods. Second, motivated by the possible inefficiencies of popular gradient-based sampling approaches in the multivariate contexts on which we focus, we introduce the simplified manifold preconditioner adaptation (SiMPA) algorithm which uses second order information about the target but avoids expensive matrix operations. We demostrate the performance and efficiency improvements of our methods relative to alternatives in extensive synthetic and real world remote sensing and community ecology applications with large scale data at up to hundreds of thousands of spatial locations and up to tens of outcomes. Software for the proposed methods is part of R package 'meshed', available on CRAN.
Inverse kinematics is an important and challenging problem in the operation of industrial manipulators. This study proposes a simple inverse kinematics calculation scheme for an industrial serial manipulator. The proposed technique can calculate appropriate values of the joint variables to realize the desired end-effector position and orientation while considering the motion costs of each joint. Two scalar functions are defined for the joint variables: one is to evaluate the end-effector position and orientation, whereas the other is to evaluate the motion efficiency of the joints. By combining the two scalar functions, the inverse kinematics calculation of the manipulator is formulated as a numerical optimization problem. Furthermore, a simple algorithm for solving the inverse kinematics via the aforementioned optimization is constructed on the basis of the simultaneous perturbation stochastic approximation with a norm-limited update vector (NLSPSA). The proposed scheme considers not only the accuracy of the position and orientation of the end-effector but also the efficiency of the robot movement. Therefore, it yields a practical result of the inverse problem. Moreover, the proposed algorithm is simple and easy to implement owing to the high calculation efficiency of NLSPSA. Finally, the effectiveness of the proposed method is verified through numerical examples using a redundant manipulator.
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