Towards prediction of low dimensional QoIs with data-driven LES closures in 2D turbulence Rik Hoekstra (rik@cwi.nl) , Wouter Edeling & Daan Crommelin Problem description Goal: ▶Predict low dimensional quantities of interest for a multiscale system. More specific: their statistical distribution. Learn a data-driven LES closure: 1 ”CNN” approach: ▶Unclosed component is a field, same size as solution. 2 ”Tau orthogonal” approach: ▶Unclosed components are time series. Why: ▶Learning time series is low dimensional and leads to explainable SGS terms. Large eddy simulation The coarse grained vorticity stream formulation of Navier-Stokes: ∂¯ω ∂t + ¯ J ( ¯ψ, ¯ω) = ν∇2¯ω + µ(¯F −¯ω) + r, (1) ∇2 ¯ψ = ¯ω, J (ψ, ω) := ∂ψ ∂x ∂ω ∂y −∂ψ ∂y ∂ω ∂x . (2) Subgrid scale (SGS) contributions are captured in: r = ¯ J ( ¯ψ, ¯ω) −J (ψ, ω). (3) For accurate LES we need a closure term: ˜r  ¯ψ, ¯ω  ≈r (ψ, ω). (4) ▶We apply: ▶a slow high fidelity solver, ▶a faster low fidelity solver which needs a closure term. ▶Parameter values for Earth sized aqua planet: ▶ν ≈4.4 · 10−6, µ ≈1.8 · 10−3. 1⃝Convolutional neural network as closure We use a CNN architecture inspired by Guan et al. [2]. The network consists of 11 2D-conv layers, with 5 × 5 kernels and periodic padding. The hidden layers have 64 channels. CNN ¯ω ¯ ψ ¯f ˜r ▶Trained on 1800 snapshots. ▶Correlation of 0.95 on test set. 2⃝Tau orthogonal method [1] Goal: track d low dimensional QoIs Qi(t) = ZZ qi(¯ω, ¯ψ)dxdy, i = 1, .., d. (5) Assume ˜r = d X i=1 τi(t)Pi(x, y, t), (6) and orthogonality ∂qi ∂¯ω, Pj = 0, if i ̸= j. (7) Now the i-th term of ˜r only influences Qi: dQi dt = resolved contributions + τi ∂qi ∂¯ω, Pi . (8) Track all Qi in the training domain using τi = dQi D ∂qi ∂¯ω, Pi E, dQi = Qref i −QLF i . (9) ▶It is possible to apply the Tau orthogonal method and the CNN simultaneously. [1] W. Edeling and D. Crommelin. Reducing data-driven dynamical subgrid scale models by physical constraints. Computers & Fluids, 201:104470, 2020. [2] Y. Guan, A. Subel, A. Chattopadhyay, and P. Hassanzadeh. Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES. Physica D: Nonlinear Phenomena, 443:133568, 2023. Overview modeling approach ω(t = 0) HF ω(t) comp. expensive ¯ω(t = 0) LF ¯ω(t) r coarse graining eq. (1) + (2) integrate QoI: RR φ(¯ω, ¯ψ)dxdy 0 250 500 750 1000 days 3.6 3.8 4.0 4.2 4.4 Energy 1e 4 Results QoIs of the form Ei,j or Zi,j, which denote the energy/enstrophy in wave numbers i ≤∥k∥≤j. With 4 QoIs: E0,15, Z0,15, E16,21 and Z16,21. Training (500 days): 0 50 100 150 200 250 0 5 1e 8 dE0, 15 0 50 100 150 200 250 days 1 0 1e 8 dE16, 21 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 E0, 15 1e 4 0.0 0.5 1.0 1.5 2.0 Density 1e4 ref LF + TO CNN + TO LF Probability density functions ▶Adding the CNN centers the time series around zero. ▶Given the time series the tracking of the QoIs is exact. Extrapolation (5000 days): ▶Using mean of dQi series for TO method 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 E0, 15 1e 4 0 1 2 Density 1e4 ref LF LF + TO CNN CNN + TO 0.6 0.8 1.0 1.2 Z0, 15 1e 2 0 2 4 Density 1e2 ref LF LF + TO CNN CNN + TO 1 2 3 4 5 E16, 21 1e 6 0 1 Density 1e6 ref LF LF + TO CNN CNN + TO 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Z16, 21 1e 3 0 2 4 Density 1e3 ref LF LF + TO CNN CNN + TO ▶The low fidelity simulation gives biased QoIs. Adding the TO method improves the distributions of E and Z in the higher wave numbers. ▶The CNN based closure works remarkably well. Adding the TO method with constant dQi does not improve it. Outlook Need better prediction for time series: ▶Fit a stochastic process to the time series. ▶Use (recurrent) neural network to predict the time series. ▶Use GAN to predict the time series. This research is funded by the Netherlands Organization for Scientific Research (NWO) through the ENW-M1 project ”Learning small closure models for large multiscale problems” CWI is the national research institute for mathematics and computer science in the Netherlands. ht