ODIL

ODIL (Optimizing a Discrete Loss) is a Python framework for solving inverse and data assimilation problems for partial differential equations. ODIL formulates the problem through optimization of a loss function including the residuals of a finite-difference and finite-volume discretization along with data and regularization terms. ODIL solves the same problems as PINN (Physics-Informed Neural Networks) but more efficiently.

Key features:

• automatic differentiation using TensorFlow or JAX
• optimization by gradient-based methods (Adam, L-BFGS) and Newton's method
• orders of magnitude lower computational cost than PINN [1]
• multigrid decomposition for faster optimization [2]

Interactive demos

These demos use a C++ implementation of ODIL with autodiff and Emscripten to run interactively in the web browser.

| <img src="https://pkarnakov.github.io/autodiff/media/wasm_poisson.png" width=120> | <img src="https://pkarnakov.github.io/autodiff/media/wasm_wave.png" width=120> | <img src="https://pkarnakov.github.io/autodiff/media/wasm_heat.png" width=120> | <img src="https://pkarnakov.github.io/autodiff/media/wasm_advection.png" width=120> | <img src="https://pkarnakov.github.io/autodiff/media/wasm_advection2.png" width=120> | |:---:|:---:|:---:|:---:|:---:| | Poisson | Wave | Heat | Advection | Advection2 |

Installation

pip install odil

or

pip install git+https://github.com/cselab/odil.git

Using uv

uv venv --python 3.12
. .venv/bin/activate
uv sync --group dev --extra tensorflow --extra jax

Using GPU

To enable GPU support, provide a non-empty list of devices in CUDA_VISIBLE_DEVICES. Values CUDA_VISIBLE_DEVICES= and CUDA_VISIBLE_DEVICES=-1 disable GPU support.

Developers

ODIL is developed by researchers at Harvard University

• Petr Karnakov
• Sergey Litvinov

advised by

• Prof. Petros Koumoutsakos

Publications

1. Karnakov P, Litvinov S, Koumoutsakos P. Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural networks. PNAS Nexus, 2024. DOI:10.1093/pnasnexus/pgae005

2. Karnakov P, Litvinov S, Koumoutsakos P. Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation. Eur. Phys. J, 2023. DOI:10.1140/epje/s10189-023-00313-7 | arXiv:2303.04679 | slides

3. Balcerak M, Amiranashvili T, Wagner A, Weidner J, Karnakov P, Paetzold JC, et al. Physics-regularized multi-modal image assimilation for braintumor localization. NeurIPS, 2024. PDF

4. Balcerak M, Weidner J, Karnakov P, Ezhov I, Litvinov S, Koumoutsakos P, et al. Individualizing glioma radiotherapy planning by optimization of a data and physics-informed discrete loss. Nature Communications, 2025. DOI:10.1038/s41467-025-60366-4

5. Buhendwa B Aaron B., Koumoutsakos P. Data-driven shape inference in three-dimensional steady-state supersonic flows: Optimizing a discrete loss with JAX-fluids. Phys Rev Fluids, 2025. DOI:10.1103/9wj9-nmr8 | arXiv:2408.10094

6. Karnakov P, Amoudruz L, Koumoutsakos P. Optimal navigation in microfluidics via the optimization of a discrete loss. Phys Rev Lett, 2025. DOI:PhysRevLett.134.044001 | arXiv:2506.15902

7. Amoudruz L, Karnakov P, Koumoutsakos P. Contactless precision steering of particles in a fluid inside a cube with rotating walls. Journal of Fluid Mechanics, 2025. DOI:10.1017/jfm.2025.10174 | arXiv:2506.15958 | Videos 1 2 3

8. Amoudruz L, Litvinov S, Papadimitriou C, Koumoutsakos P. Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss. arXiv, 2025. arXiv:2510.15664