math.NANov 3, 2026

Physics-informed traditional and generative Mori-Zwanzig-reduced models for multiscale systems

Authors: Emily Williams, Saad Qadeer, Amanda Howard, Panos Stinis

Organizations: Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, WA 99354 · Massachusetts Institute of Technology, MA 02139 · Oden Institute for Computational Engineering and Sciences, University of Texas at Austin

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

Model reduction via the Mori-Zwanzig (MZ) formalism is well understood for coarse-grained representations constructed from standard basis functions such as Fourier functions. We show that the same formalism can be used to build reduced-order models in bases discovered by physics-informed machine learning. The fluctuation-dissipation theorem (FDT) is used to close the resulting non-Markovian evolution equations for the expansion coefficients of the learned basis. We further introduce a complementary, generative closure model. Relative to the Markovian dynamics alone, both the MZ and generative closures substantially improve predictions of the filtered states.

Figures & tables

Figure 1: Technical approach of applying the MZ formalism to the system of ODEs for the physics-informed expansion coefficients.
Figure 2: Technical approach of the FDT implementation for a system of ODEs for the physics-informed expansion coefficients.
Figure 3: Collection of initial conditions from Gauss-Hermite quadrature space for Nq=2N_{q}=2 .
Figure 4: Fluctuations flk(t)f_{lk}(t) and kernels glj(t)g_{lj}(t) .
Figure 5: Memory kernels κlk(t)\kappa_{lk}(t) .
Figure 6: Full-order aa , reduced uu , and Markovian bb coefficients.
Figure 7: True filtered field s‾\overline{s} compared to Markovian and reduced approximations s^\hat{s} with absolute error fields for initial condition s0=esin⁡(x)s_{0}=e^{\sin(x)} .
Figure 8: Relative ℓ2\ell_{2} error in time.
 Average relative ℓ2\ell_{2} error
Markovian6.6%
Reduced2.9%
Table 1: Average errors for independent test sample for the viscous Burgers’ equation.
Figure 9: Technical approach of generative SGS modeling for system of ODEs for the physics-informed expansion coefficients.
Figure 10: Generated closure τkSGS\tau_{k}^{SGS} with flow and score matching.
Figure 11: Full-order aa , generated reduced rr , and Markovian bb coefficients.
Figure 12: True filtered field s‾\overline{s} compared to Markovian and reduced approximations s^\hat{s} with absolute error fields for initial condition s0=esin⁡(x)s_{0}=e^{\sin(x)} .
Figure 13: Relative ℓ2\ell_{2} error in time (black: samples, red: average).
 Average relative ℓ2\ell_{2} error
Markovian6.6%
Reduced2.4%
Table 2: Average errors for generative closure model for viscous Burgers’.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

 FlowScore
Network size3×1283\times 1284×2564\times 256
Learning rate0.0010.0001
Activation functionSiLUSiLU
Training epochs50005000
Batch size10050
Unconditional training probability0.10.2
Appendix
Table 3: Generative closure model training parameters
More in this topic

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

See all 103 papers

Explore similar work

CardsList

  1. Mamba-Assisted Non-Markovian Closure for Reduced-Order Modeling

    Jun 3, 2026Zhi-Feng Wei, Saad Qadeer, Panos StinisReduced-Order Modeling

    Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland, WA 99354, USA · Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA · Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

  2. Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

    Jul 21, 2025Alex Guo, Michael D. GrahamReduced-Order ModelingChaotic Time-Series Prediction

    Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI 53706, USA

  3. Reduced-Order Models: The Mother of World Models

    Jul 3, 2026Rajat GhoshWorld ModelsAction-Conditioned World Models

    Independent Researcher