Learning to Compress: Deep Learning for Storage-Efficient Unsteady Adjoint-Based Error Estimation

Abstract

Adjoint-based error estimation is among the most accurate methodologies for quantifying numerical errors, forming the foundation of Adaptive Mesh Refinement (AMR) strategies in turbulent flow simulations, particularly within Large Eddy Simulations (LES). However, in unsteady problems, the need to access fully time-resolved primal fields during the backward-in-time adjoint computation, along with the corresponding injected residuals, leads to prohibitive storage requirements. This thesis addresses this bottleneck by developing and assessing data-driven compression frameworks for both injected-residual and primal fields, aiming to substantially reduce storage demands while preserving the accuracy and consistency of adjoint-based error estimation.

Two machine learning–based surrogate modeling techniques, the Convolutional AutoEncoder (CAE) and the Echo State Network (ESN), were investigated to assess their ability to capture the spatial and temporal dynamics of the compressed fields, respectively. Their performance was compared against Proper Orthogonal Decomposition (POD), which served as the benchmark method. The computational framework was implemented in OpenFOAM for both primal and adjoint solvers, while the compression and reconstruction models were trained and evaluated in Python using PyTorch and ReservoirPy packages. Two test cases were considered: a smooth Manufactured Solution (MMS) of the 1D viscous Burgers’ equation, used to verify the framework and ensure consistency under controlled conditions, and a DNS-forced 1D viscous Burgers’ problem derived from Turbulent Channel Flow (TCF) data, used to evaluate the method’s robustness in representing complex, unsteady turbulent flow dynamics.

The MMS results confirmed that the framework can accurately reconstruct both the primal and injected-residual fields without altering the adjoint-based output error estimation. When the compressed residuals were used in the adjoint computation, the recovered error indicators were in close agreement with the fully resolved reference, verifying that the compression and reconstruction stages preserve consistency. Minor differences were observed only when surrogate primals were employed for adjoint solution computation, yet these deviations remained negligible, confirming the robustness of the overall formulation before applying it to more complex unsteady cases.

For the DNS-forced 1D viscous Burgers problem, representing the wall-normal velocity fluctuations of a TCF, the framework was evaluated under more realistic, nonlinear, and temporally evolving conditions. In this case, the ESN demonstrated the highest capability in reconstructing temporally varying residuals and maintaining consistent adjoint sensitivities, benefiting from its recurrent dynamics. Across all refinement levels, the time-averaged adjoint-based error indicators remained closely aligned with the reference, even at compression ratios exceeding two orders of magnitude, confirming the framework’s ability to drastically reduce storage without compromising estimation accuracy. The CAE effectively captured spatial features but showed mild instability and localized artifacts when surrogate primal reconstructions were employed for adjoint computation. While the CAE provided better reconstruction of mean values, the ESN more accurately reproduced the second-moment statistics, leading to improved representation of unsteady dynamics. The POD, used as a benchmark, yielded stable reconstructions but was limited in representing nonlinear temporal behavior and performed notably worse than the ESN when compared at equivalent compression ratios.

Overall, the results demonstrate that the proposed compression framework can reliably reduce storage demands in unsteady adjoint-based error estimation while preserving high fidelity in the output error indicators. The findings emphasize that while POD remains the most stable baseline, CAE provides strong nonlinear spatial encoding and ESN achieves the most accurate temporal reconstruction. The framework establishes a practical and scalable foundation for applying adjoint-based AMR in LES and other turbulent flow analyses where unsteady data storage remains a critical limitation.