Modeling the deformation of surfaces, particularly garments and cloth, is a foundational task across computer vision, computer graphics, and robotics. Despite significant recent advancements, a fundamental gap remains between the continuous nature of physical fabrics and the discrete structures used in digital modeling and learning systems. Traditional physics-based simulators provide high accuracy but are computationally expensive, while emerging data-driven neural surrogates have yet to fully overcome the challenges posed by this discretization gap. This thesis aims to address these challenges by introducing novel constraints, representations, and operators, establishing a comprehensive framework to bridge the gap between continuous physics and discretised neural cloths.

To this end, we identify three core challenges and propose three corresponding contributions. We first introduce GAPS, a geometry-aware, physics-based, self-supervised neural garment draping framework. GAPS enforces inextensibility through locally computed covariance-based measures while adaptively relaxing constraints in collision regions, yielding stable and realistic draping without expensive post-processing, complemented by an RBF-based skinning that improves robustness for loose garments. Second, we propose PolyFit, a continuous and differentiable surface representation based on local polynomial n-jet functions that models each surface patch with a compact set of coefficients, substantially reducing dimensionality while providing closed-form derivatives of arbitrary order. We demonstrate its effectiveness through two applications: PolySfT, a learning-free method for monocular 3D surface reconstruction, and OneFit, a self-supervised neural draping model that predicts garment deformation directly in the functional coefficient space, achieving garment-agnostic and resolution-agnostic generalization while being up to an order of magnitude faster than existing baselines. Finally, we present FNOpt, a self-supervised cloth simulation framework that addresses the resolution dependence of conventional neural simulators. By meta-learning a neural optimizer parameterized by a Fourier Neural Operator, FNOpt operates in the spectral domain and learns dynamics between function spaces, making it naturally resolution-agnostic. This enables zero-shot super-resolution: a model trained solely on coarse meshes can produce high-fidelity simulations at fine resolutions, recovering wrinkles and geometric details absent from the training data.

These technical and methodological contributions are intended to improve the robustness, fidelity, and generalization capability of neural garment modeling, and we hope they represent a meaningful step toward more generic, scalable, and efficient simulation of deformable objects.