Maneesh Kumar Singh

Maneesh Kumar Singh

We propose \emph{Consistent CutPINN}, a framework for partial differential equations posed on bounded curved domains defined implicitly by a $\C^2$ level-set function, $Ω= \{φ< 0\}$. In this paper we develop the framework for second-order elliptic problems in two dimensions. The standard PINN loss penalises the boundary mismatch in $L^2(\partialΩ)$, but $L^2(\partialΩ)$ does not control the $H^{1/2}(\partialΩ)$ trace norm that appears in the $H^1(Ω)$ energy estimate. The consistent PINN framework of Bonito et al.~\cite{bonito2025} fixes this on the unit cube $(0,1)^d$ via a Kuhn--Tucker simplicial decomposition of the flat boundary faces, but the construction relies on the affine structure of the faces and does not carry over to smooth curved boundaries. We address this gap. Specifically, (i) we introduce a discrete $H^{1/2}(\partialΩ)$ surrogate built directly from collocation points on a $\C^2$ curve, (ii) we prove a \textit{Chord-arc} norm equivalence between this surrogate and the continuous trace norm, (iii) we establish an \emph{a priori} $H^1$ error bound on cut domains, and (iv) we derive convergence rates under Besov regularity using optimal recovery theory. Numerical experiments on a disk and a non-convex flower domain confirm that the consistent loss is much more accurate than the standard PINN loss and far more robust to cut-cell configurations.

Hsin-Ying Lee, Hung-Yu Tseng, Jia-Bin Huang et al.

Image-to-image translation aims to learn the mapping between two visual domains. There are two main challenges for many applications: 1) the lack of aligned training pairs and 2) multiple possible outputs from a single input image. In this work, we present an approach based on disentangled representation for producing diverse outputs without paired training images. To achieve diversity, we propose to embed images onto two spaces: a domain-invariant content space capturing shared information across domains and a domain-specific attribute space. Our model takes the encoded content features extracted from a given input and the attribute vectors sampled from the attribute space to produce diverse outputs at test time. To handle unpaired training data, we introduce a novel cross-cycle consistency loss based on disentangled representations. Qualitative results show that our model can generate diverse and realistic images on a wide range of tasks without paired training data. For quantitative comparisons, we measure realism with user study and diversity with a perceptual distance metric. We apply the proposed model to domain adaptation and show competitive performance when compared to the state-of-the-art on the MNIST-M and the LineMod datasets.