---

TorchSymPy solves a fundamental bottleneck in computational science: evaluating parameterized integrals over massive parameter grids is prohibitively slow with traditional scalar methods.

When analyzing physics models, optics, or machine learning objectives, analytical integrals often must be computed across dense meshes containing millions of points. Standard solvers like SciPy process these sequentially on the CPU, which can take hours. TorchSymPy addresses this by bridging the gap between exact symbolic mathematics (SymPy) and parallel GPU execution (PyTorch). It takes symbolic integral expressions, performs structural optimizations, and automatically transcompiles them into native, highly optimized PyTorch kernels. Through this "compile-once, evaluate-many" architecture, you achieve zero-overhead numerical evaluation across enormous parameter sweeps—regularly executing thousands of times faster than standard numerical libraries.

Note: this module was first developed for libphysics — then split out into a standalone library to tackle generalized parallel computational bottlenecks in integration.

---

Why TorchSymPy?

When working with analytical integrals in computational physics, optics, or machine learning, researchers often hit performance bottlenecks:

1. SymPy is great for exact mathematical manipulation but is painfully slow (or completely fails) for heavy numerical grid evaluations.
2. SciPy (e.g., scipy.integrate.quad) is highly accurate but inherently sequential, single-threaded, and cannot natively leverage GPUs.
3. PyTorch thrives on massively parallel grid evaluations, but writing structural integrators by hand is tedious and error-prone.

TorchSymPy gives you the best of all worlds. You write the math symbolically in SymPy, and TorchSymPy applies automated changes-of-variables (to handle infinite domains) and structural optimizations before transpiling it into highly optimized TorchExpr kernels. These kernels can run up to 3,400x faster than SciPy by leveraging tensor-product grids and batched GPU execution.

Installation

To install the latest stable version from PyPI:

pip install torchsympy

To install from source (development):

git clone https://github.com/ibeuler/TorchSymPy.git
cd TorchSymPy
pip install -e .

Note on PyTorch: For GPU acceleration, ensure you have a CUDA-compatible torch wheel installed (e.g., torch==2.5.1+cu121).

Quickstart

The easiest path from a symbolic integral to a batched GPU evaluation:

import torch
import torchsympy
import sympy as sp

# 1. Define your integrand symbolically
x = sp.Symbol("x", real=True)
p = sp.Symbol("p", real=True)
expr = sp.Integral(sp.exp(-p * x**2), (x, -sp.oo, sp.oo))

# 2. Compile to a TorchSymPy engine
lt = torchsympy.TorchSymPy()
texpr = lt.torchify(expr)

# 3. Evaluate massively batched parameter grids on accelerators
p_grid = torch.linspace(0.5, 100.0, 10000, dtype=torch.float64, device="cuda").unsqueeze(-1)
re, im = texpr.torch_integrate_batched(
    params_values=p_grid,
    method="gauss-legendre",
    N=501,                       # Quadrature nodes
    device="cuda",               # Target accelerator
    dtype=torch.float64,         
    chunk_size_params=4096       # Safely chunk massive batches to avoid OOM
)

print(f"Real part shape: {re.shape}") # Output: torch.Size([10000])

Core Concepts: Solvers and Backends

Once you define a symbolic integration expression, TorchSymPy provides distinct evaluation paths tailored to your mathematical structure and parameter scale:

1. eval_numeric (The Smart Wrapper)

This is the recommended high-level entry point. It traverses the SymPy expression tree and detects mathematical structures that can be vastly optimized. For instance, in highly oscillatory multi-dimensional integrals (like Fresnel diffraction), it automatically factors the problem into a separable path, avoiding the catastrophic $\mathcal{O}(N^d)$ exponential blowup of tensor-product grids. It also features a shift-invariant convolution path and controls automatic mesh refinement.

2. torch_integrate_batched() (Native Batched Backend)

The workhorse for small-to-large deterministic parameter sweeps. This backend natively implements PyTorch tensor-product rules (Gauss-Legendre, Simpson). It features zero setup overhead ($\approx 1.3$ ms for evaluation) because it globally caches its quadrature grids and entirely skips dynamic object instantiation, achieving up to 100x speedups over traditional numerical wrappers on small batches. It cleanly chunks evaluations to prevent OOM limits.

3. torchquad_integrate_vectorized() (Vectorized Backend)

Delegates evaluation entirely to the external torchquad library. It uses dynamic PyTorch broadcasting to avoid creating dense parameter meshgrids in memory. However, because it dynamically instantiates IntegrationGrid objects and performs an $\mathcal{O}(N^3)$ eigenvalue solve on every call, it carries a severe $\approx 150$ ms fixed overhead. It should only be used for massive parameter grids ($\ge 10^5$ points) where this fixed penalty is amortized, or when utilizing stochastic rules like Monte Carlo in $d \ge 4$ dimensions.

Benchmarks: Speed Gains & Accuracy vs. SciPy & SymPy

TorchSymPy evaluates parameterized integrals across vast grids immensely faster than traditional methods. In our benchmark suite evaluating a parameterized Damped Cosine $\int_{0}^{\infty} e^{-x} \cos(k x) dx$, we observe huge multi-order speedups on GPUs.

The following table demonstrates the inherent trade-off between quadrature resolution ($N$) and accuracy/speed:

Execution	Time per Point	Speedup vs SciPy	Accuracy (vs Analytical)
SciPy (nquad)	1.29816 ms	1.0x	$\sim 2.41 \times 10^{-9}$
TorchSymPy (Vectorized, N=121)	0.00056 ms	2,320x	$\sim 2.07 \times 10^{-1}$ (Low N)
TorchSymPy (Batched, N=121)	0.00030 ms	4,264x	$\sim 2.07 \times 10^{-1}$ (Low N)
TorchSymPy (Vectorized, N=2001)	0.00970 ms	138x	$\sim 5.72 \times 10^{-5}$ (Medium N)
TorchSymPy (Batched, N=2001)	0.00545 ms	245x	$\sim 5.72 \times 10^{-5}$ (Medium N)
TorchSymPy (Vectorized, N=5001)	0.02904 ms	42x	$\sim 2.90 \times 10^{-9}$ (High N)
TorchSymPy (Batched, N=5001)	0.01382 ms	90x	$\sim 2.90 \times 10^{-9}$ (High N)

(Benchmarks run on an NVIDIA RTX GPU across a 10,000 parameter grid. TorchSymPy converges to parity with SciPy while remaining orders of magnitude faster at standard resolutions).

The "Hard Integrals" Problem (Experimental Analytical Check)

While TorchSymPy achieves numeric parity with SciPy for well-behaved integrals (like $\int x^{-x} dx$), evaluating conditionally convergent oscillatory integrals over infinite domains numerically pushes all quadrature engines to their breaking points.

Consider the famously difficult oscillatory integral:

$$ \int_0^\infty \frac{\sin(x)}{\sqrt{x^2 + 1}} dx $$

The true, analytical exact value (calculated symbolically via SymPy hypergeometric functions) is 0.873084. However, if we force pure numerical evaluation without symbolic reduction:

Method	Output Value	Absolute Error	Notes
SymPy (True Analytical)	0.873084	0.0	Solved symbolically via Hypergeometric functions
SymPy (Pure evalf())	-4.000000	4.873	Completely fails convergence natively
SciPy (nquad)	1.550175	0.677	Fails with IntegrationWarning (Divergent)
TorchSymPy (GaussLegendre)	-1.343219	2.216	Breaks due to mapped infinite oscillations

Takeaway: TorchSymPy provides incredible performance scaling and accurate results matching SciPy on standard mapping domains. However, for pathological integrands (like conditionally convergent oscillations at infinity), you should rely on SymPy's exact symbolic analytical integrations before attempting numerical grid sweeps.

High-Dimensional Integrals & Monte Carlo

When evaluating integrals in high dimensions ($d \ge 4$), standard tensor-product grids (like Gauss-Legendre) suffer from the curse of dimensionality. TorchSymPy natively supports MonteCarlo sampling through its vectorized backend to overcome this.

For example, evaluating an 8-Dimensional Gaussian Integral $\int_{\mathbb{R}^8} e^{-|\mathbf{x}|^2} d\mathbf{x}$ using $5.76 \times 10^6$ evaluation points:

Method (8D Gaussian)	Output Value	Analytical Truth ($\pi^4$)
TorchSymPy (MonteCarlo)	97.5273	97.4090
TorchSymPy (GaussLegendre)	89.9769	97.4090

At this dimensionality, Monte Carlo successfully approximates the integral within $\sim 0.1%$ error, while deterministic grids severely degrade given the exact same computational budget.

Running the Test Suite

pytest tests/ -v

Examples & Tutorials

Check the examples/ directory for specific physics applications and basic integration usage, including generating Wigner functions.

License

Distributed under the MIT License. See LICENSE for more information.