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Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

arXiv 1912.08177 ↗First posted 17 Dec 2019math.NA cs.LG cs.NAcited by 96

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Abstract (arXiv)

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system's governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh-Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

Key facts. Cited by 96 papers in the map (data through 14 Oct 2026). First posted on arXiv 17 Dec 2019.

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Citing papers per year: 2021: 18; 2022: 17; 2023: 19; 2024: 17; 2025: 16; 2026: 9.

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  1. 2021Modern Koopman Theory for Dynamical Systemscited by 276
  2. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 138
  3. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 89
  4. 2024Operator Learning: Algorithms and Analysiscited by 85
  5. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 58
  6. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 56
  7. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  8. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  9. 2022SVD Perspectives for Augmenting DeepONet Flexibility and Interpretabilitycited by 35
  10. 2021One-shot learning for solution operators of partial differential equationscited by 30
  11. 2022gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identificationcited by 30
  12. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 26
38 more
  1. 2021An Extensible Benchmark Suite for Learning to Simulate Physical Systemscited by 24
  2. 2023Generative learning for nonlinear dynamicscited by 22
  3. 2021Reduced operator inference for nonlinear partial differential equationscited by 21
  4. 2023Machine Learning for Partial Differential Equationscited by 21
  5. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  6. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  7. 2023Multi-fidelity reduced-order surrogate modelingcited by 18
  8. 2021Learning to Forecast Dynamical Systems from Streaming Datacited by 17
  9. 2021Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inferencecited by 16
  10. 2023Canonical and Noncanonical Hamiltonian Operator Inferencecited by 15
  11. 2023A DeepONet multi-fidelity approach for residual learning in reduced order modelingcited by 14
  12. 2024CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equationscited by 14
  13. 2024Physics-informed active learning with simultaneous weak-form latent space dynamics identificationcited by 13
  14. 2022Operator inference with roll outs for learning reduced models from scarce and low-quality datacited by 11
  15. 2023Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inferencecited by 11
  16. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  17. 2021Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learningscited by 9
  18. 2022Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learningcited by 9
  19. 2023Interpreting and generalizing deep learning in physics-based problems with functional linear modelscited by 9
  20. 2023Solving multiphysics-based inverse problems with learned surrogates and constraintscited by 9
  21. 2021Output-weighted and relative entropy loss functions for deep learning precursors of extreme eventscited by 8
  22. 2022A composable machine-learning approach for steady-state simulations on high-resolution gridscited by 8
  23. 2025Thermodynamically Consistent Latent Dynamics Identification for Parametric Systemscited by 8
  24. 2025Accelerating scientific discovery with the common task frameworkcited by 7
  25. 2023Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftingscited by 6
  26. 2024Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulationscited by 6
  27. 2024CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilationcited by 6
  28. 2021Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectoriescited by 5
  29. 2021Active operator inference for learning low-dimensional dynamical-system models from noisy datacited by 5
  30. 2022Physics-informed ConvNet: Learning Physical Field from a Shallow Neural Networkcited by 5
  31. 2022Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learningcited by 5
  32. 2022Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structurescited by 5
  33. 2023Operator learning for hyperbolic partial differential equationscited by 5
  34. 2026OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inferencecited by 5
  35. 2021LQResNet: A Deep Neural Network Architecture for Learning Dynamic Processescited by 4
  36. 2021A Latent space solver for PDE generalizationcited by 4
  37. 2022Certified data-driven physics-informed greedy auto-encoder simulatorcited by 4
  38. 2024Energy-Preserving Reduced Operator Inference for Efficient Design and Controlcited by 4

50 most cited of 96 All 96 (JSON)