A symbolic-to-MLIR codegen tool for numerical relativity. This project parses LaTeX expressions or MLIR dialect based files of spacetime metrics with a custom C++ parser and automatically generates lowered MLIR code for initial data in numerical relativity.
The project aims to build a bridge between symbolic physics and compiler technology, using MLIR to define a customizable intermediate representation for relativistic computations.
It aspires to empower physicists and researchers to write high-level tensorial expressions, which are then compiled down to optimized low-level code (LLVM IR, GPU kernels, etc.).
This is currently a proof of concept. The system extracts metric tensors from symbolic LaTeX, simplifies them, and emits valid code for use in numerical simulations of general relativity. The current options are :
- Custom LaTex parser
- Custom high level MLIR Dialect from LaTeX to create relativistic tensor patterns
- Emit lower passes MLIR (affine,Linealg/tensor/memref) from metric Tex files
- Can print Custom AST from the parser
- Parse LaTeX-formatted metrics (e.g. Schwarzschild, Kerr-Schild, FLRW)
- Generate metric MLIR dialect (still in progress)
- Lowerging metric dialect to be compiled
- Support symbolic derivation of Christoffel, Ricci, and Riemann tensors
- Provide clean initial data for BSSN-based codes
- Integrate with Tensorium for runtime execution and benchmarking
Tested on macOS 14 / MacPorts.
Adapt the paths and targets to your platform if needed.
# 1) fetch sources
git clone https://github.com/llvm/llvm-project.git ~/src/llvm-project
cd ~/src
# 2) configure an out-of-tree build directory
export PREFIX=/opt/local/libexec/llvm-20 # install location
mkdir llvm-build-rtti && cd llvm-build-rtti
cmake -G Ninja ../llvm \
-DCMAKE_INSTALL_PREFIX=$PREFIX \
-DLLVM_ENABLE_PROJECTS="mlir;clang;lld" \
-DLLVM_TARGETS_TO_BUILD="X86" \
-DLLVM_ENABLE_RTTI=ON \
-DMLIR_ENABLE_BINDINGS_PYTHON=OFF \
-DCMAKE_BUILD_TYPE=Release
# 3) build & install
ninja -j$(sysctl -n hw.logicalcpu)
sudo ninja installOnce installed, add the toolchain to your environment:
echo 'export PATH='"$PREFIX"'/bin:$PATH' >> ~/.zshrc
echo 'export DYLD_LIBRARY_PATH='"$PREFIX"'/lib:$DYLD_LIBRARY_PATH' >> ~/.zshrc
echo 'export CMAKE_PREFIX_PATH='"$PREFIX"':$CMAKE_PREFIX_PATH' >> ~/.zshrc
source ~/.zshrc
You can now configure Tensorium_MLIR:
mkdir build && cd build && cmake .. && make -jExample: spatial Schwarzschild (Kerr–Schild) and its extraction:
func.func @SchwarzschildSpatial(%x: vector<4xf64>) -> tensor<3x3xf64> {
%g = relativity.metric.get "schwarzschild_ks" {params = {M = 1.000000e+00 : f64}} %x
: vector<4xf64> -> tensor<4x4xf64>
%gamma = relativity.metric.spatial %g
: tensor<4x4xf64> -> tensor<3x3xf64>
return %gamma : tensor<3x3xf64>
}Determinant and inverse:
func.func @DetInvGamma(%x: vector<4xf64>) -> (f64, tensor<3x3xf64>) {
%gamma = call @SchwarzschildSpatial(%x)
: (vector<4xf64>) -> tensor<3x3xf64>
%det = "relativity.det3x3"(%gamma)
: (tensor<3x3xf64>) -> f64
%inv = "relativity.inv3x3"(%gamma)
: (tensor<3x3xf64>) -> tensor<3x3xf64>
return %det, %inv : f64, tensor<3x3xf64>
}Lower the Relativity dialect:
./build/bin/relativity-opt test/Relativity/test_metric_spatial.mlir --rel-expand-metric --rel-extract-spatial --rel-linalg-lower --mlir-print-op-generic -o test/Relativity/test_expand_metric_3p1.mlirLower the Relativity dialect:
➜ build git:(main) ✗ mlir-opt test/Relativity/test_expand_metric_3p1.mlir \
--convert-tensor-to-linalg \
--convert-linalg-to-loops \
--one-shot-bufferize="bufferize-function-boundaries" \
--convert-vector-to-llvm \
--convert-math-to-llvm \
--convert-arith-to-llvm \
--convert-scf-to-cf \
--convert-cf-to-llvm \
--finalize-memref-to-llvm \
--convert-func-to-llvm \
--reconcile-unrealized-casts \
> lowered.mlirTranslate to LLVM IR and compile with a C++ tester:
mlir-translate --mlir-to-llvmir lowered.mlir > lowered.ll
clang++ -c lowered.ll -o metric_tensor.o -O3 && clang++ test/Relativity/test.cpp metric_tensor.o -o test_metric -lm -O3 -std=c++17exemple Output:
Schwarzschild spatial metric at:
X = [0.0000000000000000e+00, 1.0000000000000000e+00, 1.0000000000000000e+00]
gamma (spatial metric) (sizes=[3,3], strides=[3,1])
1.3849001794597506e+00 3.8490017945975069e-01 3.8490017945975069e-01
3.8490017945975069e-01 1.3849001794597506e+00 3.8490017945975069e-01
3.8490017945975069e-01 3.8490017945975069e-01 1.3849001794597506e+00
det(gamma) = 2.1547005383792515e+00
gamma^{-1} (inverse) (sizes=[3,3], strides=[3,1])
8.2136720504591820e-01 -1.7863279495408194e-01 -1.7863279495408194e-01
-1.7863279495408194e-01 8.2136720504591820e-01 -1.7863279495408194e-01
-1.7863279495408194e-01 -1.7863279495408194e-01 8.2136720504591820e-01To convert LaTeX metric blocks into MLIR or C++ code, simply provide your .tex input to the frontend tool. The system will extract and simplify metric components, then emit either "lowered" MLIR, or dialect MLIR for the Tensorium/Relativity dialects. Status
This is currently a proof of concept. The system extracts metric tensors from symbolic LaTeX, simplifies them, and emits valid code for use in numerical simulations of general relativity.
create a test.tex file with:
$-(1 - \frac{2 M r}{\rho^2}) dt^2$
$- \frac{4 M a r \sin^2\theta}{\rho^2} dt d\phi$
$\frac{\rho^2}{\Delta} dr^2$
$\rho^2 d\theta^2$
$(r^2 + a^2 + \frac{2 M a^2 r \sin^2\theta}{\rho^2}) \sin^2\theta d\phi^2$then use tensorium-tex to parse and convert into a high level MLIR dialect with --dialect or lowered to low level MLIR with --mlir:
./tensorium-tex --dialect/mlir test.tex then you can pass it to relativity-opt :
./relativity-opt output_symbolic.mlirIf you want to lower the custom relativity dialect to LLVM IR, you can use the following command:
./bin/relativity-opt \
--relativity-simplify \
--lower-relativity \
--assemble-metric-tensor \
output_symbolic.mlir -o out.mlirmlir-opt out.mlir \
--convert-tensor-to-linalg \
--convert-linalg-to-loops \
--one-shot-bufferize="allow-return-allocs,bufferize-function-boundaries" \
--finalize-memref-to-llvm \
--convert-math-to-llvm \
--convert-arith-to-llvm \
--convert-scf-to-cf \
--convert-cf-to-llvm \
--convert-func-to-llvm \
--reconcile-unrealized-casts \
> lowered.mlir
mlir-translate --mlir-to-llvmir lowered.mlir > lowered.lldisclamer: atm the lowering pipeline is working but the metric tensor results are not accurate
bibitex :
@software{touzalin_tensorium_mlir_2025_16950267,
author = {Louis Touzalin},
title = {{Tensorium-MLIR: Relativity Dialect \& Codegen}},
version = {v0.1.0-prealpha},
year = {2025},
month = aug,
publisher = {Zenodo},
doi = {10.5281/zenodo.16950267},
url = {https://doi.org/10.5281/zenodo.16950267}
}MIT — see LICENSE