arXiv subjects

Ken Shirakawa

Publications and source records attributed to Ken Shirakawa.

At least 19 recordsLinked to original sources

arXiv2026

Large-Time Behavior of Pseudo-Parabolic Equations Associated with Generalized Total Variation Energies

In this paper, we study the well-posedness and large-time behavior of a pseudo-parabolic problem associated with a generalized total variation energy in the $BV$-framework. A main mathematical issue arises from the mismatch between the Sobolev regularity of solutions at finite times and the $BV$-structure of the corresponding steady-state problem. We prove the existence and uniqueness of solutions and investigate the relationship between their large-time behavior and solutions to the steady-state problem. Moreover, our results include the convergence of the solution trajectory to the unique steady-state solution under a typical setting arising in image processing.

math.AP↗

arXiv2026

Phantom Evidence: How and Why Generative AI Manufactures False Positives in Science

Four centuries ago Francis Bacon warned against the anticipations of nature, hasty generalization that wins assent on a few facts, and set against it the table of absence: checking that a property fails to appear where it should not. The demand was that looking convincing should not, on its own, count as evidence. Science has professed that demand ever since, while in practice letting persuasiveness do the work of evidence. It could be let to do so because making something persuasive was itself hard. Generative AI removes that difficulty, and an old error returns on a scale and at a speed it never had before. We locate the problem not in evidence growing weaker but in how surprise is counted. An observer marvels at a convincing output as a single point hit among a vast range of possibilities, yet what a system can actually reach is a small part of that range. The gap between the breadth imagined and the narrowness actually reached is what we call phantom evidence, and we formalize it as one quantity that also absorbs the trial and error and the data leakage a research process adds. Three things follow. Higher resolution and greater fluency add no evidence. The evidence a single result can carry has a ceiling that neither polishing the output nor letting a generative system grade itself can exceed. And the fraction of published findings that are true falls back to what it was before anything was observed. The prescription lies in the same place: genuinely widen what a system can reach, and measure whether convincing outputs still appear when the target is absent -- Bacon's table of absence, restated in the language of probability. In a world where the persuasive has become cheap, the credibility of science rests not on more convincing outputs but on procedures that show they could not have arisen by chance.

q-bio.NC↗

arXiv2025

Well-Posedness of Pseudo-Parabolic Gradient Systems with State-Dependent Dynamics

This paper develops a general mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics. The state dependence is induced by variable coefficient fields in the governing energy functional. Such coefficients arise naturally in scientific and technological models, including state-dependent mobilities in KWC-type grain boundary motion and variable orientation-adaptation operators in anisotropic image denoising. We establish two main results: the existence of energy-dissipating solutions, and the uniqueness and continuous dependence on initial data. The proposed framework yields a general well-posedness theory for a broad class of nonlinear evolutionary systems driven by state-dependent operators. As illustrative applications, we present an anisotropic image-denoising model and a new pseudo-parabolic KWC-type model for anisotropic grain boundary motion, and prove that both fit naturally within the abstract structure of $(\mathrm{S})_ν$.

math.AP↗

arXiv2025

Well-posedness of parabolic KWC-systems with variable-dependent mobilities

In this paper, we deal with the parabolic KWC system, associated with the mathematical model of grain boundary motion. The goal of this paper is to guarantee the well-posedness of the parabolic KWC system. However, such results have not been reported under the setting where the mobility of grain boundary motion depends on the unknown. To overcome this difficulty, results for the pseudo-parabolic type KWC system in [Antil et al., SIAM J. Math. Anal. \textbf{56}(5), 6422--6445](2024) suggest that the $H^1$-regularity of the time-derivative of the solution plays an essential role in verifying the uniqueness of the solution. In this light, we consider the pseudo-parabolic KWC system as an approximating system of the parabolic one, and focus on the improvements of regularity of solution to the parabolic system. By virtue of this regularity result, we establish the well-posedness theory on the parabolic KWC system.

math.AP↗

arXiv2025

Advancing credibility and transparency in brain-to-image reconstruction research: Reanalysis of Koide-Majima, Nishimoto, and Majima (Neural Networks, 2024)

A recent high-profile study by Koide-Majima et al. (2024) claimed a major advance in reconstructing visual imagery from brain activity using a novel variant of a generative AI-based method. However, our independent reanalysis reveals multiple methodological concerns that raise questions about the validity of their conclusions. Specifically, our evaluation demonstrates that: (1) the reconstruction results are biased by selective reporting of only the best-performing examples at multiple levels; (2) performance is artificially inflated by circular metrics that fail to reflect perceptual accuracy; (3) fair baseline comparisons reveal no discernible advantages of the study's key innovations over existing techniques; (4) the central "Bayesian" sampling component is functionally inert, producing outcomes identical to the standard optimization result; and (5) even if the component were successfully implemented, the claims of Bayesian novelty are unsubstantiated, as the proposed method does not leverage the principles of a proper Bayesian framework. These systemic issues necessitate a critical reassessment of the study's contributions. This commentary dissects these deficiencies to underscore the need for greater credibility and transparency in the rapidly advancing field of brain decoding.

q-bio.NC↗

arXiv2025

Structure-preserving scheme for 1D KWC system

In this paper, we consider a system of one-dimensional parabolic PDEs, known as the KWC system, as a phase-field model for grain boundary motion. A key feature of this system is that the equation for the crystalline orientation angle is described as a quasilinear diffusion equation with variable mobility. The goal of this paper is to establish a structure-preserving numerical scheme for the system, focusing on two main structural properties: $\sharp\,1)$ range preservation; and $\sharp\,2)$ energy dissipation. Under suitable assumptions, we construct a structure-preserving numerical scheme and address the following in the main theorems: (O) verification of the structural properties; (I) clarification of the convergence conditions; and (II) error estimate for the scheme.

math.NA↗

arXiv2025

Visual Image Reconstruction from Brain Activity via Latent Representation

Visual image reconstruction, the decoding of perceptual content from brain activity into images, has advanced significantly with the integration of deep neural networks (DNNs) and generative models. This review traces the field's evolution from early classification approaches to sophisticated reconstructions that capture detailed, subjective visual experiences, emphasizing the roles of hierarchical latent representations, compositional strategies, and modular architectures. Despite notable progress, challenges remain, such as achieving true zero-shot generalization for unseen images and accurately modeling the complex, subjective aspects of perception. We discuss the need for diverse datasets, refined evaluation metrics aligned with human perceptual judgments, and compositional representations that strengthen model robustness and generalizability. Ethical issues, including privacy, consent, and potential misuse, are underscored as critical considerations for responsible development. Visual image reconstruction offers promising insights into neural coding and enables new psychological measurements of visual experiences, with applications spanning clinical diagnostics and brain-machine interfaces.

cs.CV↗

arXiv2025

Optimal Control of Pseudo-Parabolic KWC Systems for Grain Boundary Motion

The KWC system is a well-known generic framework for phase-field models of grain boundary motion, whose original formulation is given as a parabolic gradient flow of a free energy. In the original KWC system, the results of uniqueness have been relatively scarce compared to other issues, such as existence, qualitative behavior, and numerics. This lack of progress has posed a significant challenge for more advanced topics, including optimal control. To overcome this, the authors have recently introduced the pseudo-parabolic structure to simultaneously preserve the gradient flow nature of free-energy, and to ensure the well-posedness including the uniqueness. The goal of this paper is to study an optimization problem constrained by pseudo-parabolic KWC system. The theory will be developed through a series of Main Theorems concerning the existence and semi-continuous dependence of optimal controls, and first-order necessary conditions for optimality.

math.OC↗

arXiv2025

Spurious reconstruction from brain activity

Advances in brain decoding, particularly visual image reconstruction, have sparked discussions about the societal implications and ethical considerations of neurotechnology. As these methods aim to recover visual experiences from brain activity and achieve prediction beyond training samples (zero-shot prediction), it is crucial to assess their capabilities and limitations to inform public expectations and regulations. Our case study of recent text-guided reconstruction methods, which leverage a large-scale dataset (Natural Scene Dataset, NSD) and text-to-image diffusion models, reveals limitations in their generalizability. We found poor performance when applying these methods to a different dataset designed to prevent category overlaps between training and test sets. UMAP visualization of the text features with NSD images showed a limited diversity of semantic and visual clusters, with overlap between training and test sets. Formal analysis and simulations demonstrated that clustered training samples can lead to "output dimension collapse," restricting predictable output feature dimensions. Simulations further showed that diversifying the training set improved generalizability. However, text features alone are insufficient for mapping to the visual space. We argue that recent realistic reconstructions may primarily be a blend of classification into trained categories and generation of inauthentic images through text-to-image diffusion (hallucination). Diverse datasets and compositional representations spanning the image space are essential for genuine zero-shot prediction. Interdisciplinary discussions grounded in understanding the current capabilities and limitations, as well as ethical considerations, of the technology are crucial for its responsible development.

q-bio.NC↗

arXiv2024

A gradient system based on anisotropic monochrome image processing with orientation auto-adjustment

This paper is devoted to the mathematical analysis of a system of pseudo-parabolic partial differential equations governed by an energy functional, associated with anisotropic monochrome image processing. The energy functional is based on the one proposed by [Berkels et al. Cartoon extraction based on anisotropic image classification, SFB 611, 2006], which incorporates an orientation auto-adjustment mechanism, and our energy is a simplified version which reduces the order of derivatives to improve computational efficiency. The aim of this paper is to establish a stable minimization process that addresses some instability of the algorithm caused by the reduction of derivative order. As a part of the study, we here prove Main Theorems concerned with the well-posedness of the pseudo-parabolic system, and provide a mathematically rigorous guarantee for the stability of the image denoising process derived from our system.

math.AP↗

arXiv2024

Well-posedness of a Pseudo-Parabolic KWC System in Materials Science

The original KWC-system is widely used in materials science. It was proposed in [Kobayashi et al, Physica D, 140, 141--150 (2000)] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. Firstly, it is difficult to establish its relation to physics, in particular, a variational model. Secondly, it lacks uniqueness. The former has been recently studied within the realm of BV-theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC-system. A direct relationship with variational model (as gradient-flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC-system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues.

math.AP↗

arXiv2023

A Class of Initial-Boundary Value Problems Governed by Pseudo-Parabolic Weighted Total Variation Flows

In this paper, we consider a class of initial-boundary value problems governed by pseudo-parabolic total variation flows. The principal characteristic of our problem lies in the velocity term of the diffusion flux, a feature that can bring about stronger regularity than what is found in standard parabolic PDEs. Meanwhile, our total variation flow contains singular diffusion, and this singularity may lead to a degeneration of the regularity of solution. The objective of this paper is to clarify the power balance between these conflicting effects. Consequently, we will present mathematical results concerning the well-posedness and regularity of the solution in the Main Theorems of this paper.

math.AP↗

arXiv2023

Kobayashi-Warren-Carter System of Singular Type under Dynamic Boundary Condition

In this paper, we consider a coupled system, known as Kobayashi--Warren--Carter system, abbreviated as the KWC system. KWC system consists of an Allen--Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140, 141--150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation.

math.AP↗

arXiv2023

Periodic solutions to Kobayashi--Warren--Carter systems

In this paper, a system of parabolic PDEs, called the Kobayashi--Warren--Carter system, is considered as a possible phase-field model of planar grain boundary motion. The Main Theorem is concerned with the existence of a time-periodic solution to the Kobayashi--Warren--Carter system, and the principal objective is to provide a proof without the use of a compromised assumption, which researchers have been forced to adopt in recent studies.

math.AP↗

arXiv2022

Existence of solutions to a phase-field model of 3D-grain boundary motion governed by a regularized 1-harmonic type flow

In this paper we propose a quaternion formulation for the orientation variable in the three dimensional Kobayashi--Warren model for the dynamics of polycrystals. We obtain existence of solutions to the $L^2$-gradient descent flow of the constrained energy functional via several approximating problems. In particular, we use a Ginzburg-Landau type approach and some extra regularizations. Existence of solutions to the approximating problems is shown by the use of nonlinear semigroups. Coupled with good a-priori estimates, this leads to successive passages to the limit up to finally showing existence of solutions to the proposed model. Moreover, we also obtain a maximum principle for the orientation variable.

math.AP↗

arXiv2021

Temperature Control of PDE Constrained Optimization Problems Governed by Kobayashi--Warren--Carter Type Models of Grain Boundary Motions

In this paper, we consider a class of optimal control problems governed by state-equations of Kobayashi--Warren--Carter type. The control is given by physical temperature. The focus is on problems in dimensions less than equal to 4. The results are divided in four Main Theorems, concerned with: solvability and parameter-dependence of state-equations and optimal control problems; the first order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.

math.OC↗

arXiv2021

Kobayashi--Warren--Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi--Warren--Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the $ω$-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the $ω$-limit set belong to thee space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constants solutions.

math.AP↗

arXiv2020

Optimal control problems for 1D parabolic state-systems of KWC types with dynamic boundary conditions

In this paper, we consider a class of optimal control problems governed by 1D parabolic state-systems of KWC types with dynamic boundary conditions. The state-systems are based on a phase-field model of grain boundary motion, proposed in [Kobayashi--Warren--Carter, Physica D, 140, 141--150, 2000], and in the context, the dynamic boundary conditions are supposed to reproduce the transmitted heat exchanges between interior and boundary of a polycrystal body. Our optimal control problems are labeled by using a constant $ \varepsilon \geq 0 $, and roughly summarized, the case when $ \varepsilon = 0 $ and the cases when $ \varepsilon > 0 $ correspond to the physically realistic setting, and its regularized approximating ones, respectively. Under suitable assumptions, the mathematical results concerned with: the solvability and continuous dependence for the state-systems; the solvability and $ \varepsilon $-dependence of optimal control problems; and the first order necessary optimality conditions in the problems when $ \varepsilon > 0 $ and the limiting optimality condition as $ \varepsilon \downarrow 0 $; will be obtained in forms of three Main Theorems of this paper.

math.AP↗