Machine Learning of Noise for LHD Thomson Scattering System. Keisuke Fujii, Kyoto univ.

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1 Machine Learning of Noise for LHD Thomson Scattering System Keisuke Fujii, Kyoto univ.

LHD Thomson scattering data Large helical device plasma (a) LHD-Thomson scattering system plasma r ( laser fiber array 2 3 4 5 R (m) scattered light ( Characteristics Ri0 (arb.

2 LHD Thomson scattering data Large helical device plasma (a) LHD-Thomson scattering system plasma r ( laser fiber array R (m) scattered light ( Characteristics Ri0 (arb. units) (d) 10 3 spherical mirror ~2cm, ~ points. High spatial2.0resolution: major radius (m) Steadily operated: measures almost all the LHD experiment.

3 LHD Thomson scattering data Characteristics High spatial resolution: ~2cm, ~100 points. Steadily operated: measures almost all the LHD experiment. What is required for the data analysis Inference of the smooth latent function from the discrete measurement.

4 Random noise on LHD Thomson scattering data Large noise Small noise Noise scale Uniform Gaussian noise is not appropriate. Noise scale have a dependence on latent Te or Ne values. (Diagnostic systems have a sweet spot.) Te, Ne

5 Systematic noise on LHD Thomson scattering data signal Signal mean True value Te, Ne There is systematic noise due to calibration error. (Some channels show always larger values than the vicinity.)

6 Objectives Random noise Systematic noise Latent functions signal Noise scale Signal mean True value Te, Ne Te, Ne Estimate Random noise Systematic noise Latent functions from vast amount of data.

7 Physics-driven or Data-driven? Noise scale (c) Relative sensitivity (arb. units) mirror Random noise CH5 laser wavelength CH3 CH4 CH2 CH1 CH Wavelength (nm) Physics-driven noise model List all the possible noise sources Model its distributions Propagate to the signal and merginalize (integrate) them Does not work if there is Unknown noise sources Mis-modeling of the noise property Te, Ne

8 Physics-driven or Data-driven? Uncertainty propagation usually underestimates the noise amplitude. Physics-driven noise model List all the possible noise sources Model its distributions Propagate to the signal and merginalize (integrate) them Does not work if there is Unknown noise sources Mis-modeling of the noise property Uncertainty by the diagnostic team cannot be trusted!

9 Physics-driven or Data-driven? Uncertainty propagation usually underestimates the noise amplitude. Physics-driven noise model Data-driven noise model List all the possible noise sources Directly build a noise model without considering physics Model its distributions Propagate to the signal and merginalize (integrate) them Does not work if there is Unknown noise sources Mis-modeling of the noise property Estimate these parameters from (a huge) data. Free from unknown noise sources.

10 Bayesian inference for big data Systematic noise (102x1) likelihood prior Latent functions (~102 x 105) Hyper parameters Data ~ 102 x 105 Data: Thomson scattering data (Te and Ne) for LHD experiment in Size: ~ 300,000 sets of data 1 set: ~100 radial positions x 2 kinds of values Total size: > 107 points.

11 Outlines Random noise Noise scale Systematic noise signal Signal mean True value T e, Ne Te, Ne Introduction Model Inference Result Future perspective Latent functions

12 Model : likelihood Random noise Noise scale Systematic noise Latent functions signal f exp(δy) f T e, Ne Te, Ne Student s t-distribution True value Shift by systematic noise

13 Model : prior for the calibration error δy signal f exp(δy) f Te, Ne δy p(δ y) 0 σδ Example Calibration error may distribute around 0 scale σδy θ

14 Model : noise scale model σ Neural Network 2 σte σne Te, Ne Approximation 32 2 Te Ne The noise scale dependence is approximated by N.N. (Densely connected layer)

15 Model : latent function model Latent functions Appropriate prior is not clear. We decided to determine the shape of prior from the data. We adopted low dimensional assumption for the latent functions f. 2 fte fne f (~200 points for 1 data) is described by a few parameters z (nz = 5) Prior: z ~ p(z) = N(z 0, I) 5+1 Z R

16 Outlines Random noise Noise scale Systematic noise signal Signal mean True value T e, Ne Te, Ne Introduction Model Inference Result Future perspective Latent functions

17 Optimization We need to maximize over θ [102 x 105] correction factors [102] latent parameters [5 x 105] The integral is intractable and the integrand is super-high dimensional. Variational Bayesian inference

18 Variational approximation True posterior 105 Variational posterior (factorized) [5 x 105] Optimization target: Evidence Lower Bound (ELBO) [5] [100]

19 Neural network approximation Task is now simplified; we only need to know μz, σz for each yi and μδ, σδ common for all the data. This part is still computationally expensive because large number of experimental data. Instead of evaluating μz, σz for all data separately, we constructed N.N. to directly estimate μz, σz from yi.

20 Full network structure f exp(δy) sample f= Inference network. Prediction network. Estimate μz, σz from y Estimate μy and σy from z and R.

21 Outlines Random noise Noise scale Systematic noise signal Signal mean True value T e, Ne Te, Ne Introduction Model Inference Result Future perspective Latent functions

22 Noise scale dependence Large noise Small noise Sweet spot S/N ratio get worse in very low and high Te region S/N ratio get worse in lower Ne side

23 Systematic noise Inferred calibration correction factors Systematic noise signal f exp(δy) f T e, Ne Now we have a correction factor of mis-calibration. The original data can be Post-calibrated by this values.

24 Summary Random noise We estimated Systematic noise Latent functions Random noise Systematic noise Latent functions based on the neural network variational Bayes method. With the estimated noise properties, more accurate regression becomes available.

25 Outlines Random noise Noise scale Systematic noise Latent functions signal Signal mean True value T e, Ne Te, Ne Introduction Model Inference Result Future perspective

Coordinate mapping This network can be extended to include other physical constraint. e.g. Ne and Te should be a function of magnetic flux surface.

26 Coordinate mapping This network can be extended to include other physical constraint. e.g. Ne and Te should be a function of magnetic flux surface. μ y σ y Ne Te z flux coordinate mapping ρ a0 ρ a0

27 Laplace- and variational (KL[q p]) approximation Laplace KL[q p] approximation True posterior

28 Sample from data distribution p(y) Unbiased Biased prior p(f) posteriors f

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