Recent years have witnessed tremendous progress in enabling LLMs to solve complex reasoning tasks such as math and coding. As we start to apply LLMs to harder tasks that they may not be able to solve in one shot, it is worth paying attention to their ability to construct intermediate stepping stones that prepare them to better solve the tasks. Examples of stepping stones include simplifications, alternative framings, or subproblems. We study properties and benefits of stepping stones in the context of modern reasoning LLMs via ARQ (Asking the Right Questions), a simple framework that introduces a question generator to the default reasoning pipeline. We first show that good stepping stone questions exist and are transferrable, meaning that good questions can be generated, and they substantially help LLMs of various capabilities in solving the target tasks. We next frame stepping stone generation as a post-training task and show that we can fine-tune LLMs to generate more useful stepping stones by SFT and RL on synthetic data.
This paper investigates the nonlinear dynamics of stepping flexible frames under seismic excitation. The conventional iterative method of solution of peak quasi-dynamic displacement of stepping frames is not guaranteed to converge. To address this limitation, we present closed-form solutions and stability criteria for displacement response of stepping flexible frames. Bifurcation of displacements in response of such systems is next studied through the extension of dynamics of stepping rigid bodies. An approximate analytical expression is presented to account for the effects of moving resonance under earthquake ground motions. The closed-form solutions for displacement demand can be readily adjusted to incorporate the influence of moving resonance on the quasi-dynamic response of stepping oscillators. While the quasi-dynamic method of analysis may be useful in the early stages of design, numerical integration of the nonlinear system of differential equations of motion is recommended for the solution of dynamic response in such applications. Implications for formal limit-state analysis of stepping response are discussed, accompanied by several examples demonstrating the procedures.
Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect solutions despite achieving small residual losses. This failure, we argue, is not merely an optimization difficulty. Rather, it reflects a fundamental weakness of the empirical PDE residual loss, which can admit trivial or spurious solutions during training. From this perspective, we revisit pseudo-time stepping, a technique that has recently shown strong empirical success in PINNs. We show that its main benefit is not simply to ease optimization; instead, when combined with collocation-point resampling, it helps reveal and avoid spurious solutions. At the same time, we find that the effectiveness of pseudo-time stepping depends critically on the choice of step size, which cannot be tuned reliably from the training loss alone. To overcome this limitation, we propose an adaptive pseudo-time stepping strategy that selects the step size from a finite-difference surrogate of the local residual Jacobian, yielding the largest step permitted by local sta
Humanoid robots, characterized by numerous degrees of freedom and a high center of gravity, are inherently unstable. Safe omnidirectional locomotion on stairs requires both omnidirectional terrain perception and reliable foothold selection. Existing methods often rely on forward-facing depth cameras, which create blind zones that restrict omnidirectional mobility. Furthermore, sparse post-contact unsafe stepping penalties lead to low learning efficiency and suboptimal strategies. To realize safe stair-traversal gaits, this paper introduces a single-stage training framework incorporating a dense unsafe stepping penalty that provides continuous feedback as the foot approaches a hazardous placement. To obtain stable and reliable elevation maps, we build a rolling point-cloud mapping system with spatiotemporal confidence decay and a self-protection zone mechanism, producing temporally consistent local maps. These maps are further refined by an Edge-Guided Asymmetric U-Net (EGAU), which mitigates reconstruction distortion caused by sparse LiDAR returns on stair risers. Simulation and real-robot experiments show that the proposed method achieves a near-100\% safe stepping rate on stair t
In this paper, we study the single-source shortest-path (SSSP) problem with positive edge weights, which is a notoriously hard problem in the parallel context. In practice, the $Δ$-stepping algorithm proposed by Meyer and Sanders has been widely adopted. However, $Δ$-stepping has no known worst-case bounds for general graphs. The performance of $Δ$-stepping also highly relies on the parameter $Δ$. There have also been lots of algorithms with theoretical bounds, such as Radius-stepping, but they either have no implementations available or are much slower than $Δ$-stepping in practice. We propose a stepping algorithm framework that generalizes existing algorithms such as $Δ$-stepping and Radius-stepping. The framework allows for similar analysis and implementations of all stepping algorithms. We also propose a new ADT, lazy-batched priority queue (LaB-PQ), that abstracts the semantics of the priority queue needed by the stepping algorithms. We provide two data structures for LaB-PQ, focusing on theoretical and practical efficiency, respectively. Based on the new framework and LaB-PQ, we show two new stepping algorithms, $ρ$-stepping and $Δ^*$-stepping, that are simple, with non-trivi
The athermal quasistatic deformation method provides an elegant solution to overcome the limitation of short time spans in molecular simulations. It provides overdamped conditions, allowing for the extraction of purely structural responses in the absence of thermal vibration. However, it requires computationally expensive sequences of affine deformation followed by minimization of the potential energy to incrementally find the path in the potential energy landscape that corresponds to the correct solution trajectory. Therefore, we propose an athermal parallel stepping scheme that significantly improves the computational time necessary to find the correct solution trajectory using a multi-thread approach. Our approach proposes stepping at two levels. Level I stepping provides a sequence of initial guesses at large increments by affine deformation of the system and land-marking anchor points on the potential energy landscape. Level II stepping performs a set of individual finely resolved athermal quasistatic deformation steps between the inherent structures of the initial level I guesses executed in parallel. The evaluated candidate trajectory is then verified by consecutively compar
In this Letter, local time stepping (LTS) is incorporated into the unified gas-kinetic wave-particle (UGKWP) method for steady multiscale flow simulation. It accelerates convergence step by a factor of $3.8\times$--$20\times$ and reduces wall-clock time by up to $21\times$ relative to global time stepping (GTS). A rigorous analysis of the particle flux under LTS identifies that fixed per-cell as $Δt_i$ is a sufficient condition for the time-averaged flux balance. This condition has not been stated in prior particle-based LTS work, where $Δt_i$ varies in time and the flux balance is therefore not guaranteed. Together with proportional rescaling of particle mass and free transport time at cell interfaces, the fixed-$Δt_i$ condition yields a conservative framework with no free parameters. The UGKWP-LTS method is validated on cylinder and flat-plate benchmarks that possess multiscale flow features.
Algebraic steppers help students learn functional programming by displaying evaluation as a sequence of small-step reductions, but even simple programs produce long traces in which key ideas are buried under mundane reductions. This paper presents the filtered stepper calculus, a formal framework that gives users scoped, pattern-based control over which reduction steps are shown or hidden. Users annotate programs with lightweight filter expressions that match on the structure of redexes. Filters compose via lexical scoping so that inner filters override outer ones. We prove preservation, progress, and a simulation theorem establishing that the filtered stepper agrees with the underlying unfiltered semantics, and mechanize all proofs in Agda. We implement the calculus in the Hazel live programming environment, including its support for stepping programs with holes and type errors. To do so, we reconcile Hazel's internal environment-based evaluator with the substitution-based presentation expected in the classroom. We deploy the system in a university programming languages course. Our evaluation shows that students adopt the stepper organically, though more advanced uses of filters m
Non-uniform time stepping in acoustic propagation models can be used to preserve accuracy or reduce computational cost for an acoustic simulation with a wave front propagating through a domain with both heterogeneous and homogenous regions, such as for a simulation of breast ultrasound tomography. The k-space correction already exist within the literature to remove numerical dispersion caused by the time stepping procedure in pseudo-spectral time domain models, but requires a uniform time step. Here we expand this correction to be able to account for a non-uniform time stepping method and illustrate the potential advantages and considerations. A version of this Article has been submitted for review to the Journal of Theoretical and Computational Acoustics.
We revisit adaptive time stepping, one of the classical topics of numerical analysis and computational engineering. While widely used in application and subject of many theoretical works, a complete understanding is still missing. Apart from special cases, there does not exist a complete theory that shows how to choose the time steps such that convergence towards the exact solution is guaranteed with the optimal convergence rate. In this work, we use recent advances in adaptive mesh refinement to propose an adaptive time stepping algorithm that is mathematically guaranteed to be optimal in the sense that it achieves the best possible convergence of the error with respect to the number of time steps, and it can be implemented using a time stepping scheme as a black box.
The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation, especially on parallel and hybrid supercomputers. In this paper, we propose an exponential time integration method for the Cahn-Hilliard equation and describe its efficient Krylov subspace based implementation. We compare the method to a Chebyshev polynomial local iteration modified (LIM) time stepping scheme. Both methods are explicit (i.e., do not involve linear system solution) and tested with both constant and adaptively chosen time steps.
The Dirac-Frenkel variational principle is a widely used building block for using nonlinear parametrizations in the context of model reduction and numerically solving partial differential equations; however, it typically leads to time-dependent least-squares problems that are poorly conditioned. This work introduces a randomized time stepping scheme that solves at each time step a low-dimensional, random projection of the parameter vector via sketching. The sketching has a regularization effect that leads to better conditioned least-squares problems and at the same time reduces the number of unknowns that need to be solved for at each time step. Numerical experiments with benchmark examples demonstrate that randomized time stepping via sketching achieves competitive accuracy and outperforms standard regularization in terms of runtime efficiency.
Modulation-based imaging (MoBI) is an X-ray phase-contrast technique that uses an intensity modulator (or membrane) in the beam. Although MoBI can be performed in a single shot, multiple exposures are typically needed to improve the quality of the result. The membrane is typically moved using a regular stepping pattern for convenience; however, the impact of the membrane movement scheme on image quality has not been fully investigated yet. In this work, we explore optimized movement strategies aiming at improving MoBI performance. An experimental study tested optimization schemes based on global and local standard deviation metrics, and compared them with regular and random stepping motions. The results demonstrated superior contrast-to-noise ratio and reduced angular sensitivity in the optimized approaches compared to conventional stepping. These results were consistent across different membrane types, with honeycomb membranes showing the highest compatibility with the optimization procedure. Noise power spectrum analysis further validated the advantages of the optimized motion strategies. Overall, the results demonstrate that an optimized membrane movement can significantly impro
The robust balancing capability of humanoids is essential for mobility in real environments. Many studies focus on implementing human-inspired ankle, hip, and stepping strategies to achieve human-level balance. In this paper, a robust balance control framework for humanoids is proposed. Firstly, a Model Predictive Control (MPC) framework is proposed for Capture Point (CP) tracking control, enabling the integration of ankle, hip, and stepping strategies within a single framework. Additionally, a variable weighting method is introduced that adjusts the weighting parameters of the Centroidal Angular Momentum damping control. Secondly, a hierarchical structure of the MPC and a stepping controller was proposed, allowing for the step time optimization. The robust balancing performance of the proposed method is validated through simulations and real robot experiments. Furthermore, a superior balancing performance is demonstrated compared to a state-of-the-art Quadratic Programming-based CP controller that employs the ankle, hip, and stepping strategies.
This paper deals with the problem of simulating dense dispersed systems composed by large numbers of particles undergoing ballistic aggregation. The most classical approaches for dealing with such problems are represented by the so-called event-driven methods. Despite being more accurate, these methods become computationally very expensive as the number of particles increases. Typically, their computational cost is proportional to the square of the number of particles and thus they become extremely demanding as soon as this number becomes sufficiently large. An alternative approach, called time-stepping, consists in evolving the problem over small time-intervals and to handle all collisions occurring during each time interval simultaneously. In this work, we follow this second direction and we introduce a new time stepping method which recasts the problem of the multiple collisions in a minimization framework. The objective of this work is twofold, first to show that the statistical description of the resulting aggregates obtained with this new time stepping method is sufficiently close to that of the event driven methods. The second goal consists in showing that the computational
The compactness lemma in programming language theory states that any recursive function can be simulated by a finite unrolling of the function. One important use case it has is in the logical relations proof technique for proving properties of typed programs, such as strong normalization. The relation between recursive functions and their finite counterparts is a special variant of the class of bisimulation relations. However, standard bisimulation proof approaches do not apply to the compactness lemma as properties of the relation vary over execution. As a result, the proof of compactness is often messy because the multiple copies made of the recursive function during execution can be unrolled an inconsistent number of times. We present a new proof technique by indexing the bisimulation relation over the step transitions and utilizing an intermediate "pattern" language to mechanize bookkeeping. This generalization of "pattern stepping bisimulation" obviates the need for contextual approximation within the compactness lemma, and thus extends the compactness lemma to a wider range of programming languages, including those that incorporate control flow effects. We demonstrate this ap
In legged locomotion, the relationship between different gait behaviors and energy consumption must consider the full-body dynamics and the robot control as a whole, which cannot be captured by simple models. This work studies the robot dynamics and whole-body optimal control as a coupled system to investigate energy consumption during balance recovery. We developed a 2-phase nonlinear optimization pipeline for dynamic stepping, which generates reachability maps showing complex energy-stepping relations. We optimize gait parameters to search all reachable locations and quantify the energy cost during dynamic transitions, which allows studying the relationship between energy consumption and stepping locations given different initial conditions. We found that to achieve efficient actuation, the stepping location and timing can have simple approximations close to the underlying optimality. Despite the complexity of this nonlinear process, we show that near-minimal effort stepping locations fall within a region of attractions, rather than a narrow solution space suggested by a simple model. This provides new insights into the non-uniqueness of near-optimal solutions in robot motion pla
Performing highly agile dynamic motions, such as jumping or running on uneven stepping stones has remained a challenging problem in legged robot locomotion. This paper presents a framework that combines trajectory optimization and model predictive control to perform robust and consecutive jumping on stepping stones. In our approach, we first utilize trajectory optimization based on full-nonlinear dynamics of the robot to generate periodic jumping trajectories for various jumping distances. A jumping controller based on a model predictive control is then designed for realizing smooth jumping transitions, enabling the robot to achieve continuous jumps on stepping stones. Thanks to the incorporation of MPC as a real-time feedback controller, the proposed framework is also validated to be robust to uneven platforms with unknown height perturbations and model uncertainty on the robot dynamics.
In this work, we propose a learning approach for 3D dynamic bipedal walking when footsteps are constrained to stepping stones. While recent work has shown progress on this problem, real-world demonstrations have been limited to relatively simple open-loop, perception-free scenarios. Our main contribution is a more advanced learning approach that enables real-world demonstrations, using the Cassie robot, of closed-loop dynamic walking over moderately difficult stepping-stone patterns. Our approach first uses reinforcement learning (RL) in simulation to train a controller that maps footstep commands onto joint actions without any reference motion information. We then learn a model of that controller's capabilities, which enables prediction of feasible footsteps given the robot's current dynamic state. The resulting controller and model are then integrated with a real-time overhead camera system for detecting stepping stone locations. For evaluation, we develop a benchmark set of stepping stone patterns, which are used to test performance in both simulation and the real world. Overall, we demonstrate that sim-to-real learning is extremely promising for enabling dynamic locomotion over
Photonic stepped-frequency (SF) radar offers high range resolution and only requires low-speed driving electronics, but existing architectures face challenges in achieving low phase noise and uniform frequency steps simultaneously. Here, we demonstrate a photonic SF radar system that exploits dual Brillouin lasers in a shared fiber cavity to simultaneously suppress phase noise and ensure uniform frequency stepping. Phase noise is reduced through Brillouin optomechanical suppression and common-mode noise rejection upon photomixing. Frequency-step uniformity is enforced via lasing at a series of uniformly spaced cavity resonances. The system generates an X-band SF waveform spanning 1.31 GHz, achieving >23 dB of phase-noise improvement at a 100 kHz offset relative to a low-cost driving voltage-controlled oscillator. The demonstrated system reduces the dependence of the output waveform quality on noise in the driving electronics, offering a path towards high-performance radar sensing.