Publications

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically construct the solution landscape to find all steady-state solutions connected to a homogeneous state. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability, and reveal a general nonlinear mechanism that increases the parameter space over which pattern formation occurs in the RD systems. Furthermore, by using a generalized action in the functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.

We explore a quaternion adjugate matrix-based representation for rotational motion in the Perspective-n-Point (PnP) problem. Leveraging quadratic quaternion terms within a Determinant Ratio Matrix (DRaM) estimation framework, we extend its application to perspective scenarios, providing a robust and efficient initialization for iterative PnP pose estimation. Notably, by solving the orthographic projection least-squares problem, DRaM provides a reliable initialization that enhances the accuracy and stability of iterative PnP solvers. Experiments on synthetic and real data demonstrate its efficiency, accuracy, and robustness, particularly under high noise conditions. Furthermore, our non-minimal formulation ensures numerical stability, making it effective for real-world applications.

We explore a quaternion adjugate matrix-based representation for rotational motion in the Perspective-n-Point (PnP) problem. Leveraging quadratic quaternion terms within a Determinant Ratio Matrix (DRaM) estimation framework, we extend its application to perspective scenarios, providing a robust and efficient initialization for iterative PnP pose estimation. Notably, by solving the orthographic projection least-squares problem, DRaM provides a reliable initialization that enhances the accuracy and stability of iterative PnP solvers. Experiments on synthetic and real data demonstrate its efficiency, accuracy, and robustness, particularly under high noise conditions. Furthermore, our non-minimal formulation ensures numerical stability, making it effective for real-world applications.

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand whose precise form derives directly from the theory of perfect plasticity behaves quadratically close to the origin and grows linearly once a speci c threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector eld the Cauchy stress in the terminology of perfect plasticity which allows us to de ne characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study [5], we show that this vector eld is actually continuous, save for possibly two points. The di erent behaviors of the energy density at zero and at innity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

The human auditory cortex is topographically organized. Neurons with similar response properties are spatially clustered, forming smooth maps for acoustic features such as frequency in early auditory areas, and modular regions selective for music and speech in higher-order cortex. Yet, evaluations for current computational models of auditory perception do not measure whether such topographic structure is present in a candidate model. Here, we show that cortical topography is not present in the previous best-performing models at predicting human auditory fMRI responses. To encourage the emergence of topographic organization, we adapt a cortical wiring-constraint loss originally designed for visual perception. The new class of topographic auditory models, TopoAudio, are trained to classify speech, and environmental sounds from cochleagram inputs, with an added constraint that nearby units on a 2D cortical sheet develop similar tuning. Despite these additional constraints, TopoAudio achieves high accuracy on benchmark tasks comparable to the unconstrained non-topographic baseline models. Further, TopoAudio predicts the fMRI responses in the brain as well as standard models, but unlike standard models, TopoAudio develops smooth, topographic maps for tonotopy and amplitude modulation (common properties of early auditory representation, as well as clustered response modules for music and speech (higher-order selectivity observed in the human auditory cortex). TopoAudio is the first end-to-end biologically grounded auditory model to exhibit emergent topography, and our results emphasize that a wiring-length constraint can serve as a general-purpose regularization tool to achieve biologically aligned representations.

As LLMs are increasingly deployed as agents, agentic reasoning - the ability to combine tool use, especially search, and reasoning - becomes a critical skill. However, it is hard to disentangle agentic reasoning when evaluated in complex environments and tasks. Current agent benchmarks often mix agentic reasoning with challenging math reasoning, expert-level knowledge, and other advanced capabilities. To fill this gap, we build a novel benchmark, GSM-Agent, where an LLM agent is required to solve grade-school-level reasoning problems, but is only presented with the question in the prompt without the premises that contain the necessary information to solve the task, and needs to proactively collect that information using tools. Although the original tasks are grade-school math problems, we observe that even frontier models like GPT-5 only achieve 67% accuracy. To understand and analyze the agentic reasoning patterns, we propose the concept of agentic reasoning graph: cluster the environment's document embeddings into nodes, and map each tool call to its nearest node to build a reasoning path. Surprisingly, we identify that the ability to revisit a previously visited node, widely taken as a crucial pattern in static reasoning, is often missing for agentic reasoning for many models. Based on the insight, we propose a tool-augmented test-time scaling method to improve LLM's agentic reasoning performance by adding tools to encourage models to revisit. We expect our benchmark and the agentic reasoning framework to aid future studies of understanding and pushing the boundaries of agentic reasoning.

We analyze and develop numerical methods for time-harmonic wave scattering in metallic waveguide structures of infinite extent. We show that radiation boundary conditions formulated via projectors onto outgoing modes determine the coefficients of propagating modes uniquely, even when the structure supports trapped modes. Building on this, we introduce a fast divide-and-conquer solver that constructs solution operators on subdomains as impedance-to-impedance maps and couples them by enforcing continuity conditions across their interfaces. For Dirichlet waveguides, the computation of impedance-to-impedance maps requires the solution of mixed Dirichlet-Impedance boundary value problems. We construct a second-kind Fredholm integral equation that avoids near-hypersingular operators, requiring only integral operators whose kernels are at most weakly singular. Numerical experiments on large structures with many circuit elements demonstrate substantial efficiency gains: the proposed approach typically outperforms state-of-the-art fast iterative and fast direct solvers by one to two orders of magnitude.

Classical Ewald methods for Coulomb and Stokes interactions rely on ``kernel-splitting," using decompositions based on Gaussians to divide the resulting potential into a near field and a far field component. Here, we show that a more efficient splitting for the scalar biharmonic Green's function can be derived using zeroth-order prolate spheroidal wave functions (PSWFs), which in turn yields new efficient splittings for the Stokeslet, stresslet, and elastic kernels, since these Green's tensors can all be derived from the biharmonic kernel. This benefits all fast summation methods based on kernel splitting, including FFT-based Ewald summation methods, that are suitable for uniform point distributions, and DMK-based methods that allow for nonuniform point distributions. The DMK (dual-space multilevel kernel-splitting) algorithm we develop here is fast, adaptive, and linear-scaling, both in free space and in a periodic cube. We demonstrate its performance with numerical examples in two and three dimensions.

The inclusion of symmetries as an inductive bias, known as equivariance, often improves generalization on geometric data (e.g. grids, sets, and graphs). However, equivariant architectures are usually highly constrained, designed for symmetries chosen a priori, and not applicable to datasets with other symmetries. This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly. In this work, we build a single model --- the Any-Subgroup Equivariant Network (ASEN) --- that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully permutation-equivariant base model, and then obtain subgroup equivariance by using a symmetry-breaking input whose automorphism group is that subgroup. However, finding an input with the desired automorphism group is computationally hard. We overcome this by relaxing from exact to approximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our subgroup-equivariant networks can simulate equivariant MLPs, and their universality can be guaranteed if the base model is universal. Empirically, we validate our method on symmetry selection, multitask, and transfer learning settings, demonstrating that a single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model.

Wnt-signaling, via β-catenin or the planar cell polarity (PCP) branch, is crucial for development, tissue homeostasis, and linked to many diseases. LRP5/6, arrow (arr) in Drosophila, is the obligate co-receptor in Wnt/β-catenin signaling, with ligand binding to a Frizzled (Fz) family member and LRP5/6 mediating formation of the signalosome complex with Dishevelled (Dsh/Dvl in mammals) and Axin. Current models for Wnt/PCP signaling omit Arr/LRP5/6 and the notion is that it functions without these co-receptors. Here we show that arr/LRP5/6 is positively required in Wnt/PCP signaling. In Drosophila, loss of arr results in PCP mediated cellular orientation defects, aberrant wing hair formation, and loss of polarity, as described for core PCP factors fz, fmi/Celsr, and dsh. In the eye, arr mutant tissue displays cell fate changes in photoreceptors R3/R4 and chirality defects, classical PCP phenotypes. During Wnt/PCP establishment, defects are manifest as reduced levels of Fmi/Celsr and Dsh along with loss of their asymmetric localization. Functional interactions indicate that Fz can recruit Arr, and this potentiates Fz and Dsh function in PCP signaling in all tissues tested. Taken together, our data support an essential Arr/LRP5/6 function in promoting Wnt/Fz-Dsh PCP-complex activity.