ML in Physical Phenomena

Physical simulation helps engineers understand fluid flow, heat transfer, and other coupled processes, but high-resolution calculations can be expensive. At MAIL, we combine machine learning with numerical modeling to accelerate prediction, reconstruct unresolved details, and discover interpretable governing equations. Our research connects three complementary goals:

1. Fast prediction: Graph neural networks and neural operators learn from simulation or experimental data to approximate system behavior across geometries, initial conditions, and physical parameters. We study their accuracy and ability to generalize beyond the training examples.

2. Scientific discovery: Symbolic regression identifies compact equations from observations. Projects such as LLM-SR connect language-model scientific knowledge with numerical fitting and evaluation, while LLM-SRBench provides a framework for testing equation-discovery methods.

3. Better use of simulation data: Physics-informed reconstruction and super-resolution recover fine-scale structure from coarse observations. These methods complement numerical solvers, with attention to physical consistency and errors in unresolved dynamics.

Our research at MAIL

Graph Neural Nets

Graph neural network discretizes the PDE domain as a graph and can flexibly adapt to a wide array of physical systems. Based on graph representation, we develop neural-network-based surrogate model for the following systems:

a) Graph neural network-accelerated Lagrangian fluid simulation

b) Graph convolutional networks applied to unstructured flow field data

c) Graph neural networks accelerated molecular dynamics

Neural Operators

Neural operators learn mappings between functions, such as a PDE’s input conditions and its solution field. Our Operator Transformer (OFormer), introduced in 2022 and published in TMLR in 2023, established a foundation for the lab’s transformer-based operator learning: self-attention encodes input information and cross-attention connects it to prediction locations, accommodating different sampling patterns and grids.

FactFormer (NeurIPS 2023) addresses scalability. Its axial factorized attention reduces the cost of modeling multidimensional fields, supporting high-resolution 2D flow and 3D smoke simulations. These complementary contributions connect flexible spatial representations with efficient PDE surrogate modeling.

a) Transformer for Partial Differential Equations’ Operator Learning — OFormer, TMLR 2023. Code.

b) Scalable Transformer for PDE Surrogate Modeling — FactFormer, NeurIPS 2023. Code.

Symbolic Representation

Symbolic models describe physical systems through compact, interpretable equations. We combine data-driven search with scientific knowledge to discover relationships that can be inspected and tested beyond the training data. LLM-SR (ICLR 2025) uses language models to propose equation structures, then fits and evaluates them numerically. LLM-SRBench (ICML 2025) extends this effort with a benchmark for evaluating scientific equation discovery. These projects build on our work in symbolic regression and governing-equation identification:

a) Identification of parametric dynamical systems using integer programming

b) Data-driven identification of 2D Partial Differential Equations using extracted physical features

c) Transformer-based Planning for Symbolic Regression

Super-Resolution

Machine learning can be used to learn and interpolate between the scale that’s not resolved by the discretization. Inspired by the success of deep-learning based image super-resolution techniques, we develop learning-based models to restore and upsample under-resolved simulation data:

a) A physics-informed diffusion model for high-fidelity flow field reconstruction

b) TPU-GAN: Learning temporal coherence from dynamic point cloud sequences

c) Deep learning for efficient reconstruction of high-resolution turbulent dns data

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