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Nikola Kovachki

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6 papers

1 author row

JMLR Journal 2025 Journal Article

Geometric Operator Learning with Optimal Transport

Xinyi Li
Zongyi Li
Nikola Kovachki
Anima Anandkumar

We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries. Classical geometric learning methods typically represent domains as meshes, graphs, or point clouds. Our approach generalizes discretized meshes to mesh density functions, formulating geometry embedding as an OT problem that maps these functions to a uniform density in a reference space. Compared to previous methods relying on interpolation or shared deformation, our OT-based method employs instance-dependent deformation, offering enhanced flexibility and effectiveness. For 3D simulations focused on surfaces, our OT-based neural operator embeds the surface geometry into a 2D parameterized latent space. By performing computations directly on this 2D representation of the surface manifold, it achieves significant computational efficiency gains compared to volumetric simulation. Experiments with Reynolds-averaged Navier-Stokes equations (RANS) on the ShapeNet-Car and DrivAerNet-Car datasets show that our method achieves better accuracy and also reduces computational expenses in terms of both time and memory usage compared to existing machine learning models. Additionally, our model demonstrates significantly improved accuracy on the FlowBench dataset, underscoring the benefits of employing instance-dependent deformation for datasets with highly variable geometries. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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NeurIPS Conference 2023 Conference Paper

Geometry-Informed Neural Operator for Large-Scale 3D PDEs

Zongyi Li
Nikola Kovachki
Chris Choy
Boyi Li
Jean Kossaifi
Shourya Otta
Mohammad Amin Nabian
Maximilian Stadler

We propose the geometry-informed neural operator (GINO), a highly efficient approach for learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function (SDF) representation of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. We provide an efficient implementation of GINO using an optimized hashing approach, which allows efficient learning in a shared, compressed latent space with reduced computation and memory costs. GINO is discretization-invariant, meaning the trained model can be applied to arbitrary discretizations of the continuous domain and applies to any shape or resolution. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a 26, 000x speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

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JMLR Journal 2023 Journal Article

Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs

Nikola Kovachki
Zongyi Li
Burigede Liu
Kamyar Azizzadenesheli
Kaushik Bhattacharya
Andrew Stuart
Anima Anandkumar

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

Learning Chaotic Dynamics in Dissipative Systems

Zongyi Li
Miguel Liu-Schiaffini
Nikola Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Kaushik Bhattacharya
Andrew Stuart
Anima Anandkumar

Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to $5000$.

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JMLR Journal 2021 Journal Article

On Universal Approximation and Error Bounds for Fourier Neural Operators

Nikola Kovachki
Samuel Lanthaler
Siddhartha Mishra

Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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NeurIPS Conference 2020 Conference Paper

Multipole Graph Neural Operator for Parametric Partial Differential Equations

Zongyi Li
Nikola Kovachki
Kamyar Azizzadenesheli
Burigede Liu
Andrew Stuart
Kaushik Bhattacharya
Anima Anandkumar

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

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