A PyTorch module for differentiable parametric curves with learnable coefficients, such as a B-Spline curve with learnable control points.

Fully differentiable curve implementations that integrate seamlessly with PyTorch's autograd system. It streamlines use cases such as continuous numerical embeddings for embedding-based models (e.g. factorization machines [6] or transformers [2,3]), Kolmogorov-Arnold networks [1], or path planning in robotics.

Docs

• Documentation site.
• Example notebooks for you to try our

Features

• Fully Differentiable: Custom autograd function ensures gradients flow properly through the curve evaluation.
• Batch Processing: Vectorized operations for efficient batch evaluation.

Installation

pip install torchcurves

uv add torchcurves

Use cases

There are examples in the examples directory showing how to build models using this library. Here we show some simple code snippets to appreciate the library.

Use case 1 - continuous embeddings

import torchcurves as tc
from torch import nn
import torch


def Net(nn.Module):
    def __init__(self, num_categorical, num_numerical, dim, num_knots=10):
        super().__init__()
        self.cat_emb = nn.Embedding(num_categorical, dim)
        self.num_emb = tc.BSplineCurve(num_numerical, dim, knots_config=num_knots)
        self.my_super_duper_transformer = MySuperDuperTransformer()

    def forward(self, x_categorical, x_numerical):
        embeddings = torch.cat([self.cat_emb(x_categorical), self.num_emb(x_numerical)], axis=-2)
        return self.my_super_duper_transformer(embeddings)

Use case 2 - Kolmogorov-Arnold networks

A KAN [1] based on the B-Spline basis, along the lines of the original paper:

import torchcurves as tc
from torch import nn

input_dim = 2
intermediate_dim = 5
num_control_points = 10

kan = nn.Sequential(
    # layer 1
    tc.BSplineCurve(input_dim, intermediate_dim, knots_config=num_control_points),
    tc.Sum(dim=-2),
    # layer 2
    tc.BSplineCurve(intermediate_dim, intermediate_dim, knots_config=num_control_points),
    tc.Sum(dim=-2),
    # layer 3
    tc.BSplineCurve(intermediate_dim, 1, knots_config=num_control_points),
    tc.Sum(dim=-2),
)

Yes, we know the original KAN paper used a different curve parametrization, B-Spline + arcsinh, but the whole point of this repo is showing that KAN activations can be parametrized in arbitrary ways.

For example, here is a KAN based on Legendre polynomials of degree 5:

import torchcurves as tc
from torch import nn

input_dim = 2
intermediate_dim = 5
degree = 5

kan = nn.Sequential(
    # layer 1
    tc.LegendreCurve(input_dim, intermediate_dim, degree=degree),
    tc.Sum(dim=-2),
    # layer 2
    tc.LegendreCurve(intermediate_dim, intermediate_dim, degree=degree),
    tc.Sum(dim=-2),
    # layer 3
    tc.LegendreCurve(intermediate_dim, 1, degree=degree),
    tc.Sum(dim=-2),
)

Since KANs are the primary use case for the tc.Sum() layer, we can omit the dim=-2 argument, but it is provided here for clarity.

Advanced features

The curves we provide here typically rely on their inputs to lie in a compact interval, typically [-1, 1]. Arbitrary inputs need to be normalized to this interval. We provide two simple out-of-the-box normalization strategies described below.

Rational scaling

This is the default strategy — this strategy computes

$$x \to \frac{x}{\sqrt{s^2 + x^2}},$$

and is based on the paper

Wang, Z.Q. and Guo, B.Y., 2004. Modified Legendre rational spectral method for the whole line. Journal of Computational Mathematics, pp.457-474.

In Python it looks like this:

tc.BSplineCurve(curve_dim, normalization_fn='rational', normalization_scale=s)

Arctan scaling

This strategy computes

$$x \to \frac{2}{\pi} \arctan(x / s).$$

This kind of scaling function, up to constants, is the CDF of the Cauchy distribution. It is useful when our inputs are assumed to be heavy tailed.

In Python it looks like this:

tc.BSplineCurve(curve_dim, normalization_fn='arctan', normalization_scale=s)

Clamping

The inputs are simply clipped to [-1, 1] after scaling, i.e.

$$x \to \max(\min(1, x / s), -1)$$

In Python it looks like this:

tc.BSplineCurve(curve_dim, normalization_fn='clamp', normalization_scale=s)

Custom normalization

Provide a custom function that maps its input to the designated range after scaling. Example:

def erf_clamp(x: Tensor, scale: float = 1, out_min: float = -1, out_max: float = 1) -> Tensor:
    mapped = torch.special.erf(x / scale)
    return ((mapped + 1) * (out_max - out_min)) / 2 + out_min

tc.BSplineCurve(curve_dim, normalization_fn=erf_clamp, normalization_scale=s)

Example: B-Spline KAN with clamping

A KAN based on rationally scaled B-Spline basis with the default scale of $s=1$:

spline_kan = nn.Sequential(
    # layer 1
    tc.BSplineCurve(input_dim, intermediate_dim, knots_config=knots, normalization_fn='clamp'),
    tc.Sum(),
    # layer 2
    tc.BSplineCurve(intermediate_dim, intermediate_dim, knots_config=knots, normalization_fn='clamp'),
    tc.Sum(),
    # layer 3
    tc.BSplineCurve(intermediate_dim, 1, knots_config=knots, normalization_fn='clamp'),
    tc.Sum(),
)

Legendre KAN with rational clamping

import torchcurves as tc
from torch import nn

input_dim = 2
intermediate_dim = 5
degree = 5

config = dict(degree=degree, normalization_fn="clamp")
kan = nn.Sequential(
    # layer 1
    tc.LegendreCurve(input_dim, intermediate_dim, **config),
    tc.Sum(),
    # layer 2
    tc.LegendreCurve(intermediate_dim, intermediate_dim, **config),
    tc.Sum(),
    # layer 3
    tc.LegendreCurve(intermediate_dim, 1, **config),
    tc.Sum(),
)

Development

Development Installation

Using uv (recommended):

# Clone the repository
git clone https://github.com/alexshtf/torchcurves.git
cd torchcurves

# Create virtual environment and install
uv venv
uv sync --all-groups

Running Tests

# Run all tests
uv run pytest

# Run with coverage
uv run pytest --cov=torchcurves

# Run specific test file
uv run pytest tests/test_bspline.py -v

Performance Benchmarks

This project includes opt-in performance benchmarks (forward and backward passes) using pytest-benchmark.

Location: benchmarks/

Run benchmarks:

# Run all benchmarks
uv run pytest benchmarks -q

# Or select only perf-marked tests if you mix them into tests/
uv run pytest -m perf -q

CUDA timing notes: We synchronize before/after timed regions for accurate GPU timings.

Compare runs and fail CI on regressions:

# Save a baseline
uv run pytest benchmarks --benchmark-save=legendre_baseline

# Compare current run to baseline (fail if mean slower by 10% or more)
uv run pytest benchmarks --benchmark-compare --benchmark-compare-fail=mean:10%

Export results:

uv run pytest benchmarks --benchmark-json=bench.json

Building the docs

# Prepare API docs
cd docs
make html

Citation

If you use this package in your research, please cite:

@software{torchcurves,
  author = {Shtoff, Alex},
  title = {torchcurves: Differentiable Parametric Curves in PyTorch},
  year = {2025},
  publisher = {GitHub},
  url = {https://github.com/alexshtf/torchcurves}
}

References

[1]: Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljacic, Thomas Y. Hou, Max Tegmark. "KAN: Kolmogorov–Arnold Networks." ICLR (2025).
[2]: Juergen Schmidhuber. "Learning to control fast-weight memories: An alternative to dynamic recurrent networks." Neural Computation, 4(1), pp.131-139. (1992)
[3]: Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. "Attention is all you need." Advances in neural information processing systems 30 (2017).
[4]: Alex Shtoff, Elie Abboud, Rotem Stram, and Oren Somekh. "Function Basis Encoding of Numerical Features in Factorization Machines." Transactions on Machine Learning Research.
[5]: Rügamer, David. "Scalable Higher-Order Tensor Product Spline Models." In International Conference on Artificial Intelligence and Statistics, pp. 1-9. PMLR, 2024.
[6]: Steffen Rendle. "Factorization machines." In 2010 IEEE International conference on data mining, pp. 995-1000. IEEE, 2010.