💻 computer science

SharpNet: Enhancing MLPs to Represent Functions with Controlled Non-differentiability

SharpNet is a modified MLP architecture that enables precise representation of functions with controlled non-differentiability by augmenting the network with an auxiliary feature function derived from Poisson's equation, thereby allowing the joint optimization of feature locations and network parameters to accurately recover sharp edges and corners without blurring.

Original authors: Hanting Niu, Junkai Deng, Fei Hou, Wencheng Wang, Ying He

Published 2026-07-07

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CC BY 4.0

Original authors: Hanting Niu, Junkai Deng, Fei Hou, Wencheng Wang, Ying He

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to draw a perfect 3D object, like a toy robot or a piece of furniture. You want the computer to learn the smooth curves of the robot's arm, but you also need it to learn the sharp, crisp edges where the arm meets the body.

The Problem: The "Too Smooth" Artist
Standard AI tools used for this job (called MLPs) are like artists who only know how to use a soft, round brush. They are amazing at painting smooth, flowing landscapes. But if you ask them to draw a sharp corner, they tend to "smudge" it. They try to make the transition so smooth that the sharp edge turns into a rounded bump. This is a problem for things like CAD (computer-aided design) where sharp edges are essential.

Other methods try to fix this by chopping the object into tiny puzzle pieces (patches) or using a grid of pixels. But this is like trying to draw a sharp corner using only square Lego bricks; you either get a jagged, pixelated edge, or you can't draw an open edge at all because the puzzle pieces don't fit together right.

The Solution: SharpNet
The authors created a new tool called SharpNet. Think of SharpNet as giving the artist a special "stencil" or a "guide rail" that tells them exactly where to stop smoothing and start making a sharp cut.

Here is how it works, using a few simple analogies:

The "Invisible Fence" (The Feature Function):
Imagine you want to draw a sharp line on a piece of paper. Instead of just guessing where the line goes, you lay down a thin, invisible fence.
- On one side of the fence, the paint flows smoothly.
- On the other side, the paint flows smoothly too.
- But right at the fence, the paint suddenly changes direction.
SharpNet creates this "fence" mathematically. It uses a special formula (based on something called Poisson's equation) to create a field that is perfectly smooth everywhere except exactly where the user wants the sharp edge. This ensures the AI knows exactly where to make the "cut."
The "Smart Brush" (The Network):
The AI (the MLP) looks at this invisible fence. It learns to say, "Okay, I will be smooth here, but right at this fence, I will change my behavior instantly." Because the math behind the fence is very precise, the AI doesn't accidentally make sharp edges in the wrong places (like the jagged artifacts seen in other methods).
Learning the Fence (The Magic Trick):
Usually, you have to tell the computer exactly where the sharp edges are. But SharpNet is clever. If you don't know where the edges are, the AI can actually learn to move the fence itself. It adjusts the position of the "invisible fence" while it learns the shape, finding the perfect spot to make the sharp edge. It's like a sculptor who not only carves the stone but also figures out exactly where the chisel should strike to get the sharpest corner.

Why is this better?

No Smudging: Unlike standard AI, SharpNet keeps the sharp edges sharp and the smooth parts smooth.
No Puzzle Pieces: It doesn't need to chop the object into tiny patches. It can handle complex shapes, open edges, and weird curves without getting confused.
Flexible: It works whether you give it a perfect 3D model to copy, or just a messy cloud of points (like a laser scan) and ask it to figure out the shape.

Real-World Tests
The authors tested SharpNet on:

2D Shapes: Drawing lines and curves where the "sharpness" happens at specific points. SharpNet got it right; others made it blurry or jagged.
3D CAD Models: Reconstructing complex machines and toys. SharpNet successfully recreated sharp corners and edges that other methods either rounded off or broke apart.

In Summary
SharpNet is a new way for computers to draw 3D shapes. It solves the problem of "too smooth" AI by adding a mathematical guide that tells the AI exactly where to create sharp, crisp edges, while keeping the rest of the object perfectly smooth. It's like giving the computer a ruler and a sharp knife, so it can draw both curves and corners with perfect precision.

Technical Summary: SharpNet

Problem Statement
Multi-layer perceptrons (MLPs) are standard tools for function approximation and geometry representation (e.g., Neural Signed Distance Functions in NeRFs and SDFs). However, standard MLPs with smooth activation functions produce globally smooth ( $C^\infty$ ) outputs, while ReLU-based MLPs are only piecewise linear ( $C^0$ ) but lack precise control over the location of non-differentiability. This limitation makes it difficult for existing methods to represent functions that are continuous but possess intentional gradient discontinuities (sharp edges, corners, or medial axes) at specific, user-defined, or learnable locations. Existing approaches, such as grid-based methods (InstantNGP) or patch-subdivision methods (NH-Rep), often introduce artifacts, grid-aligned artifacts, or fail to handle open (non-closed) sharp features and complex topologies.

Methodology
The authors propose SharpNet, an MLP architecture designed to represent functions that are smooth almost everywhere but exhibit controlled non-differentiability ( $C^0$ continuity with gradient jumps) at prescribed locations.

Feature Function via Poisson's Equation:
SharpNet augments the standard MLP input with an auxiliary feature function, $f: \Omega \to \mathbb{R}$ . This function is defined as the solution to Poisson's equation with jump Neumann boundary conditions on a feature set $M$ (a curve in 2D or surface in 3D):
- $\nabla^2 f(x) = h(x)$ for $x \in \Omega \setminus M$ .
- Continuity condition: $f$ is continuous across $M$ .
- Jump condition: The normal derivative of $f$ exhibits a jump discontinuity across $M$ .
Using Green's third identity, the solution is expressed as a boundary integral involving the Green's function of the Laplacian. For computational efficiency, the authors simplify this by setting $h(x)=0$ and the jump magnitude to 1, reducing the feature function to an integral of the Green's function over the feature set $M$ .
Localization and Acceleration:
To avoid the high cost of global integration, the authors introduce a mollifier (a smooth, compactly supported function). This allows the global integral to be decomposed into a sum of local integrals. The feature function is computed as a weighted sum of local contributions, ensuring the function remains $C^0$ on $M$ and $C^\infty$ elsewhere. This formulation is fully differentiable with respect to the geometry of $M$ , enabling joint optimization.
Network Architecture and Composition:
The final network is defined as $\Phi_\theta(x, f(x))$ , where $x$ are spatial coordinates and $f(x)$ is the auxiliary feature. Theoretical analysis (Theorem 3.1) confirms that if the network is locally sensitive to the feature channel (i.e., $\partial \Phi / \partial f \neq 0$ ), the non-differentiability of $f$ is transferred to the output.

To handle complex geometries where the magnitude of the gradient jump changes abruptly (e.g., at junctions of sharp edges), the feature set $M$ is partitioned into disjoint subsets $\{M^{(i)}\}$ . Each subset is associated with a distinct feature function $f^{(i)}$ . The network input concatenates these multiple feature functions, allowing the model to represent piecewise-varying directional derivative jumps.
Optimization:
The framework supports two modes:
- Fixed Features: When the feature geometry $M$ is known (e.g., from a mesh), $f$ is treated as a fixed prior.
- Learnable Features: When $M$ is unknown, the parameters defining the geometry of $M$ are optimized jointly with the MLP weights $\theta$ . The differentiability of the feature function with respect to $M$ enables end-to-end learning of sharp feature locations.

Key Contributions

Controlled Non-differentiability: SharpNet provides a principled way to enforce $C^0$ continuity with gradient discontinuities at specific, user-specified, or learnable locations, overcoming the global smoothness limitation of standard MLPs.
Meshless Formulation: Unlike patch-based or grid-based methods, SharpNet does not require domain subdivision or mesh partitioning. It can represent open (non-closed) sharp edges and complex topologies without the topological constraints of CSG trees or patch intersections.
Learnable Feature Geometry: The method introduces a differentiable feature function based on Poisson's equation, allowing the sharp feature locations themselves to be learned from data (e.g., point clouds) without requiring pre-segmentation.
Efficient Evaluation: The use of mollifiers and closed-form Green's function integrals on line segments/triangles enables efficient local evaluation, making the approach scalable for 3D reconstruction.

Results
The authors validate SharpNet on 2D distance field fitting and 3D CAD reconstruction tasks:

2D Experiments: In geodesic distance fitting and medial axis learning, SharpNet (using Softplus activation) accurately recovers sharp transitions and medial axes. It outperforms standard MLPs (which oversmooth), ReLU-based MLPs (which introduce spurious creases), and InstantNGP (which produces grid-aligned artifacts).
3D CAD Reconstruction:
- From Meshes: SharpNet reconstructs models with sharp edges more accurately than NH-Rep, particularly for open sharp curves and high-genus shapes where NH-Rep fails due to patch subdivision limitations.
- From Oriented Points: SharpNet surpasses InstantNGP and SIREN in Chamfer Distance (CD), Hausdorff Distance (HD), and Normal Error (NE), effectively recovering sharp edges without grid artifacts.
- From Raw Points: In the absence of normal supervision, SharpNet outperforms NeurCADRecon and SIREN, demonstrating robustness in learning both geometry and sharp features from unoriented point clouds.
Quantitative Metrics: Across all datasets (100 CAD models), SharpNet consistently achieves lower CD, HD, and NE, and higher F-1 scores compared to state-of-the-art baselines.

Significance and Claims
The paper claims that SharpNet offers a flexible and precise alternative to existing neural field representations for geometry with sharp features. By leveraging classical PDE theory (Poisson's equation with jump Neumann conditions) and Green's functions, it bridges the gap between smooth neural approximations and the need for sharp geometric fidelity. The authors emphasize that their method avoids the "leaking" of sharpness to unintended regions (a problem in mesh-based or grid-based interpolation) and supports the joint learning of feature geometry, making it applicable to scenarios where sharp features are not known a priori. The work positions SharpNet as a superior backbone for implicit CAD reconstruction, capable of handling complex, open, and high-genus structures that challenge current patch-based or grid-based approaches.

Technical Summary: SharpNet

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