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Decodable and Sample Invariant Continuous Object Encoder
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Proposes HDFE, a training-free encoder that maps samples of a Lipschitz continuous object into a decodable, sample-invariant, distance-preserving complex vector, and applies it to PDE solving and point-cloud normal estimation.
Problem
Encoding continuous objects (functions, implicit surfaces) from sparse, irregularly sampled points into fixed-length vectors that are invariant to sampling distribution and density, while remaining decodable so neural networks can take and predict continuous objects.
Approach
Superpose hyper-dimensional encodings of samples, with binding of input (fractional power encoding of x) and output (encoding of y), weighted by an iterative refinement that converges to the center of the smallest enclosing ball of sample encodings; this gives sample invariance, decodability by similarity search, and isometry without any training.
Key results
- On PCPNet, replacing PointNet with HDFE reduces average normal RMSE from 14.58 to 13.67 (0.91 improvement), and with data augmentation to 12.88; on FamousShape the average drops from 24.95 to 22.59 and 21.16 with augmentation.
- Integrating HDFE into HSurf-Net lowers average normal RMSE from 10.18 to 10.07 on PCPNet and from 17.74 to 17.68 on FamousShape; with augmentation 9.93 and 17.44.
- For PDE solving, HDFE reportedly achieves about 55% lower prediction error than MGKN and PCANN and competitive performance with FNO, while outperforming FNO when decoding is not required.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace PointNet with HDFE | PCPNet · PCPNet | average RMSE: 14.58 -> 13.67 | Helped |
| Replace PointNet with HDFE | PCPNet · FamousShape | average RMSE: 24.95 -> 22.59 | Helped |
| Integrate HDFE into HSurf-Net | HSurf-Net · PCPNet | average RMSE: 10.18 -> 10.07 | Helped |
| Integrate HDFE into HSurf-Net | HSurf-Net · FamousShape | average RMSE: 17.74 -> 17.68 | Helped |
| Add data augmentation to HDFE (ablation) | PCPNet - PointNet + HDFE · PCPNet | average RMSE: 13.67 -> 12.88 | Helped |
| Add data augmentation to HDFE (ablation) | PCPNet - PointNet + HDFE · FamousShape | average RMSE: 22.59 -> 21.16 | Helped |
| Add data augmentation to HDFE (ablation) | HSurf-Net + HDFE · PCPNet | average RMSE: 10.07 -> 9.93 | Helped |
| Add data augmentation to HDFE (ablation) | HSurf-Net + HDFE · FamousShape | average RMSE: 17.68 -> 17.44 | Helped |
| Replace PointNet with HDFE | PCPNet · PCPNet Density-Gradient | RMSE: 13.46 -> 10.19 | Helped |
| Replace PointNet with HDFE | PCPNet · FamousShape Density-Gradient | RMSE: 19.50 -> 14.55 | Helped |
| Integrate HDFE into HSurf-Net | HSurf-Net · PCPNet Density-Gradient | RMSE: 5.03 -> 4.87 | Helped |
| Integrate HDFE into HSurf-Net | HSurf-Net · FamousShape Density-Gradient | RMSE: 7.44 -> 7.35 | Helped |
Surprising findings
- A PointNet trained as a function encoder with a decoder fails to produce reasonable reconstructions, even after 700 epochs, motivating a training-free approach.
Limitations
- Limited encoding capacity: underfitting for functions over large domains or highly nonlinear functions.
- Reconstruction error dominates PDE error in some settings and requires larger embedding dimension.
- Gains on the SOTA HSurf-Net are small (0.11 and 0.06 average RMSE improvements without augmentation).
Experimental setup
Datasets: 1D Burgers' equation benchmark 2D Darcy Flow benchmark PCPNet FamousShape
Benchmarks: Burgers 1D (PDE) Darcy Flow 2D (PDE) PCPNet FamousShape
Metrics: prediction error (absolute distance) root mean squared angle error (RMSE)
Baselines: FNO MGKN PCANN PCPNet HSurf-Net PointNet
Builds on: fractional power encoding hyper-dimensional computing vector function architecture PointNet Fourier neural operator
Closest papers by meaning
- Unsupervised Inference of Signed Distance Functions From Single Sparse Point Clouds Without Learning Priors (CVPR 2023 · similarity 0.84)
- Higher-Order Function Networks for Learning Composable 3D Object Representations (ICLR 2020 · similarity 0.84)
- HSurf-Net: Normal Estimation for 3D Point Clouds by Learning Hyper Surfaces (NeurIPS 2022 · similarity 0.84)
- Inferring Neural Signed Distance Functions by Overfitting on Single Noisy Point Clouds through Finetuning Data-Driven based Priors (NeurIPS 2024 · similarity 0.83)
- Invaria: Learning Scale and Density Invariance in Point Clouds via Next-Resolution Prediction (NeurIPS 2026 · similarity 0.83)
- Learning Bijective Surface Parameterization for Inferring Signed Distance Functions from Sparse Point Clouds with Grid Deformation (CVPR 2025 · similarity 0.83)
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