arXiv subjects

Meenakshi Krishnan

Publications and source records attributed to Meenakshi Krishnan.

4 recordsLinked to original sources

arXiv2026

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

cs.LG↗

arXiv2025

ViscoReg: Neural Signed Distance Functions via Viscosity Solutions

Implicit Neural Representations (INRs) that learn Signed Distance Functions (SDFs) from point cloud data represent the state-of-the-art for geometrically accurate 3D scene reconstruction. However, training these Neural SDFs often requires enforcing the Eikonal equation, an ill-posed equation that also leads to unstable gradient flows. Numerical Eikonal solvers have relied on viscosity approaches for regularization and stability. Motivated by this well-established theory, we introduce ViscoReg, a novel regularizer that provably stabilizes Neural SDF training. Empirically, ViscoReg outperforms state-of-the-art approaches such as SIREN, DiGS, and StEik on ShapeNet, the Surface Reconstruction Benchmark, and 3D scene reconstruction datasets. Additionally, we establish novel generalization error estimates for Neural SDFs in terms of the training error, using the theory of viscosity solutions.

cs.GR↗

arXiv2025

Error analysis for learning the time-stepping operator of evolutionary PDEs

Deep neural networks (DNNs) have recently emerged as effective tools for approximating solution operators of partial differential equations (PDEs) including evolutionary problems. Classical numerical solvers for such PDEs often face challenges of balancing stability constraints and the high computational cost of iterative solvers. In contrast, DNNs offer a data-driven alternative through direct learning of time-stepping operators to achieve this balancing goal. In this work, we provide a rigorous theoretical framework for analyzing the approximation of these operators using feedforward neural networks (FNNs). We derive explicit error estimates that characterize the dependence of the approximation error on the network architecture -- namely its width and depth -- as well as the number of training samples. Furthermore, we establish Lipschitz continuity properties of time-stepping operators associated with classical numerical schemes and identify low-complexity structures inherent in these operators for several classes of PDEs, including reaction-diffusion equations, parabolic equations with external forcing, and scalar conservation laws. Leveraging these structural insights, we obtain generalization bounds that demonstrate efficient learnability without incurring the curse of dimensionality. Finally, we extend our analysis from single-input operator learning to a general multi-input setting, thereby broadening the applicability of our results.

math.NA↗

arXiv2025

3D Gaussian Splatting with Normal Information for Mesh Extraction and Improved Rendering

Differentiable 3D Gaussian splatting has emerged as an efficient and flexible rendering technique for representing complex scenes from a collection of 2D views and enabling high-quality real-time novel-view synthesis. However, its reliance on photometric losses can lead to imprecisely reconstructed geometry and extracted meshes, especially in regions with high curvature or fine detail. We propose a novel regularization method using the gradients of a signed distance function estimated from the Gaussians, to improve the quality of rendering while also extracting a surface mesh. The regularizing normal supervision facilitates better rendering and mesh reconstruction, which is crucial for downstream applications in video generation, animation, AR-VR and gaming. We demonstrate the effectiveness of our approach on datasets such as Mip-NeRF360, Tanks and Temples, and Deep-Blending. Our method scores higher on photorealism metrics compared to other mesh extracting rendering methods without compromising mesh quality.

cs.GR↗