11 citations · 20 across the 14 of their papers we have counts for
14 papers
Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks
Kazuaki Tanaka, Kohei Yatabe
The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as…
A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions
Kazuaki Tanaka, Ryoga Iwanami, Kaname Matsue +1
This paper presents a novel framework for enclosing solutions of Poisson's equation based on generalized sub- and super-solutions constructed using fundamental solutions. The conve…
Polygonal Sequence-driven Triangulation Validator: An Incremental Approach to 2D Triangulation Verification
Sora Sawai, Kazuaki Tanaka, Katsuhisa Ozaki +1
Two-dimensional Delaunay triangulation is a fundamental aspect of computational geometry. This paper presents a novel algorithm that is specifically designed to ensure the correctn…
Numerical verification method for positive solutions of elliptic problems
Kazuaki Tanaka
The purpose of this paper is to propose methods for verifying the positivity of a weak solution of an elliptic problem assuming -error estimation $ \left\|u-\hat{u}\…
A posteriori verification of the positivity of solutions to elliptic boundary value problems
Kazuaki Tanaka, Taisei Asai
The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither -regula…
Numerical verification for asymmetric solutions of the Hénon equation on bounded domains
Taisei Asai, Kazuaki Tanaka, Shin'ichi Oishi
The Hénon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore,…
A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations
Kazuaki Tanaka
This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogen…
Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems
Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi
In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We revea…
Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains
Makoto Mizuguchi, Kazuaki Tanaka, Kouta Sekine +1
This paper is concerned with an explicit value of the embedding constant from to for a bounded domain , where $1\leq…
Numerical verification method for positiveness of solutions to elliptic equations
Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi
In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic equations. We provide a sufficient condition for a solution to an el…
Rigorous numerical enclosures for positive solutions of Lane-Emden's equation with sub-square exponents
Kazuaki Tanaka, Michael Plum, Kouta Sekine +2
The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation with homogeneous Dirichlet boundary conditions. We pro…
Numerical validation of blow-up solutions of ordinary differential equations
Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki +3
This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on c…
Sharp numerical inclusion of the best constant for embedding on bounded convex domain
Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi +1
In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding on bounded conv…
Estimation of the Sobolev embedding constant on domains with minimally smooth boundary
Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi +1
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing…