Physics-Informed Recurrent Neural Networks for...
Transcript of Physics-Informed Recurrent Neural Networks for...
Physics-Informed Recurrent Neural Networksfor Land Model Surrogate Construction
Vishagan Ratnaswamy 1 Cosmin Safta 1 Khachik Sargsyan 1
Daniel M. Ricciuto 2
1 Sandia National Laboratories, Livermore CA
2Oak Ridge National Laboratory, Oak Ridge TN
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 1 / 28
Acknowledgements
DOE, Office of Science,
I Advanced Scientific Computing Research (ASCR)I Scientific Discovery through Advanced Computing (SciDAC) programI Biological and Environmental Research (BER)I National Energy Research Scientific Computing Center (NERSC)
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and EngineeringSolutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s
National Nuclear Security Administration under contract DE-NA-0003525.
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 2 / 28
Outline
MotivationI Energy Exascale Earth System Model (E3SM): Land component
Need for Surrogate ModelsI Polynomial Chaos SurrogateI Neural Network Surrogates
F Multilayer PerceptronF LSTM RNNF Tree-LSTM RNN
Global Sensitivity Analysis (GSA)
Preliminary Results
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 3 / 28
E3SM Model Overview
Energy Exascale Earth System Model (E3SM) is a coupled earth model
US Department of Energy (DOE) sponsored Earth system model https://e3sm.org
Ocean, Atmosphere, Sea ice and Land Components
Computationally expensive to run the coupled mode (including ocean and atmosphere)
Can only do a few global simulations of coupled E3SM models → hard to get training data
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 4 / 28
E3SM Land Model (ELM) Overview
Land model incorporates a set of biogeophysical processes:
Cheapest component, can run in single column modeSimplified python model available (sELM)Can evaluate model many times for various input parameters
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 5 / 28
ELM Produces Time Series givenInput Parameters and Forcing Drivers
Daily Forcings (site dependent)1 Min/Max Temperature2 Day of Year3 Solar radiation4 Water Availability
Thirty year record (1980-2009)of forcings
47D Stochastic Space
QOIs dependent on previousday’s output
initializemodel
Inputs Drivers
20.00 20.25 20.50 20.75 21.00 21.25 21.50 21.75 22.00year since 1980
0
50
100
150
200
250
300
350
FSDS
US-Ha1US-Ha1+0.5 latUS-Ha1-0.5 lat
−2 −1 0 1 2 30
0.2
0.4
0.6
0.8
1
QOI
160 180 200 220 240 260days
5
0
5
10
15
20
GPP
dataMLPLSTMTree-LSTM
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 6 / 28
Surrogates are necessary for expensive computationalmodels
... otherwise called supervised ML
Surrogates are required for ensemble-intensive studies, such asI parameter estimationI uncertainty propagationI global sensitivity analysisI optimal experimental design
This talk
Investigating surrogate construction approaches for ELMto enable all of the above
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 7 / 28
Polynomial chaos (PC) surrogate for black-box f (λ)
Represent QOIs as orthogonal expansion of random variables
f (λ) ≈ fc(λ(ξ)) =∑α∈I
cαΨα(ξ)
Germ: ξ = (ξ1, ξ2, ..., ξd), e.g. uniform: λ(ξ) is a linear scaling
Multi-index α = {α1, . . . , αd}Orthogonal polynomials wrt p(ξ), Ψα(ξ) =
∏di=1 ψαi (ξi )
Typical construction approach: regression to find PC modes cαAdvantages of PC
I moment estimation, uncertainty propagation, global sensitivity analysisExpensive model challenge
I Use Bayesian regression, helps to quantify lack of simulation dataHigh-d challenge d � 1
I Number of terms in expansion of order p and dimension d : |I| = (d+p)!d!p!
I Use sparse regression
We employ Bayesian compressed sensing (BCS): iterative algorithmfor Bayesian sparse learning [Babacan, 2010; Sargsyan, 2014; Ricciuto, 2018]
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 8 / 28
‘Free’ Global Sensitivity Analysis with PC f (λ(ξ)) '∑α∈I
cαΨα(ξ)
Main effect sensitivity indices
Si =Var [E(f (λ|λi )]
Var [f (λ)]=
∑α∈Ii
c2α||Ψα||2∑α∈I\{0}
c2α||Ψα||2
I Ii is the set of bases with only ξi involvedI Si is the uncertainty contribution that is due to i-th parameter only
Total effect sensitivity indices
Ti = 1− Var [E(f (λ|λ−i )]
Var [f (λ)]=
∑α∈ITi
c2α||Ψα||2∑α∈I\{0}
c2α||Ψα||2
I ITi is the set of bases with ξi involved, including all its interactions.I Ti is the total uncertainty contribution due to i-th parameter
[Sudret, 2008; Crestaux, 2009; Sargsyan, 2017]
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 9 / 28
Surrogate Modeling: MLP Approach
Benign feed-forward networkDoes not account for temporal aspect of model
I Cannot propagate information of QOIs day to day (or month to month)
Model dealt with as a black box
……
……
x1
x2
x3
x47
……
……
……
……
……
……
………………..
Hidden Layer 1 Hidden Layer 2 Hidden Layer N-1
Daily Forcing
Stochastic Input
GPPNEELAINPP
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 10 / 28
Surrogate Modeling: LSTM-RNN
Vanilla LSTM Recurrent NN architecture
Accounts for temporal aspect of model
Model dealt with as a black box
Day 1 Day 2 Day 3
…………
Day N
x1 x2 x3
…………
x47
Daily Forcing
Stochastic Input
GPPNEELAINPP
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 11 / 28
Graph Structure of ELM Land ModelLooking under the hood helps build physics-informed architecture
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 12 / 28
Surrogate Modeling: Tree LSTM RNN
Retain similar structure to Gated RNN ([Tai, 2015])
Incorporate physical connections into LSTM
Each QOI has a LSTM gate
GPP
LAI
QOIDay 1GPP
LAI
QOIDay 2
…………x1 x2 x3 x47
NEE
NPP
NEE
NPP
LSTM RNNVanilla
GPP
LAI
QOIDay 1GPP
LAI
QOIDay 2
…………x1 x2 x3 x47
NEE
NPP
NEE
NPP
Tree-LSTM RNNPhysics-informed
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 13 / 28
For NN surrogates, GSA can be carried out withMonte-Carlo
Mean estimate: E [f (λ)] ≈ 1N
N∑n=1
f (λ(n))
Variance estimate: Var [f (λ)] ≈ 1N
N∑n=1
f (λ(n))2 − E [f (λ)]2
Main sensitivity:
Si =Var [E(f (λ|λi )]
Var [f (λ)]≈ 1
Var [f (λ)]
(1
N
N∑n=1
f (λ(n))f (λ′(n)−i ∪ λ
(n)i )− E [f (λ)]2
),
where λ′(n)−i ∪ λ
(n)i is a single-column swap sample given two sampling
schemes λ(n) and λ′(n)
... similar estimators for total sensitivity
Inherits all the challenges of Monte-Carlo
[Jansen, 1999; Sobol, 2001; Saltelli, 2002]
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 14 / 28
Overview of Surrogate Models Implemented
Surrogatemodels
ML
LSTM-RNN
GSA
Tree-LSTMRNN
PCE
GSA
• Di�cult withtime depend.
• Easy tocompute
• Need MC
• No timedepend.
• No hierarchy
• Hierarchy
MLP
Surrogates also needed for inverse UQ (calibration)A
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 15 / 28
Summary of Case Study
Training Details
Generate samples from sELM model: 30 years (1980-2009)
Each training set (time history) has 10, 950 data points (daily)
Simulation at University of Michigan Biological Station site
500 training samples, 500 validation samples
NN Details
Train on daily QoIs
500 Epochs of SGD
2 layers, 150 neurons
L2 loss, dropout regularization
MLP
No time dynamics
No physics
LSTM
Time dynamics
No physics
Tree-LSTM
Time dynamics
Physics
PC Details
Hard to train on dailyaverages (noisy)
Train on monthly averages
Use Bayesian compressivesensing to computecoefficients
Build surrogate for eachaverage month, i.e. 30× 12surrogates
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 16 / 28
Snapshot of training sample 1
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 17 / 28
Snapshot of GPP training sample 2
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 18 / 28
Snapshot of GPP training sample 3
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 19 / 28
Snapshot of GPP validation sample 1
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 20 / 28
Snapshot of GPP validation sample 2
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 21 / 28
Snapshot of GPP validation sample 3
160 180 200 220 240 260day
2
4
6
8
10
12
14
16
GPP
MLPLSTMTree-LSTMdata
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 22 / 28
GPP Parity Plot comparisons for various months
5 0 5 10 15sELM
5
0
5
10
15Su
rroga
te
March
MLPLSTMPCETree-LSTM
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 23 / 28
GPP Parity Plot comparisons for various months
5 0 5 10 15sELM
5
0
5
10
15Su
rroga
te
August
MLPLSTMPCETree-LSTM
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 24 / 28
GPP Parity Plot comparisons for various months
5 0 5 10 15sELM
5
0
5
10
15Su
rroga
te
December
MLPLSTMPCETree-LSTM
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 25 / 28
Global Sensitivity Analysis Comparison
gdd_
crit
crit_
dayl
nday
s_on
nday
s_of
fnu
e_tre
enu
e_gr
ass
slato
p_ev
erg
slato
p_de
cidliv
ewdc
nle
afcn
froot
cnfs
tor2
tran
stem
_leaf
croo
t_st
emf_
livew
dfro
ot_le
afrg
_fra
cbr
_mr
q10_
mr
csto
r_ta
ur_
mor
tlw
top_
ann
leaf
_long
froot
_long
q10_
hrk_
l1k_
l2k_
l3k_
s1k_
s2k_
s3k_
s4k_
frag
rf_l1
s1rf_
l2s2
rf_l3
s3rf_
s1s2
rf_s2
s3rf_
s3s4
soil4
cicw
d_fli
gfr_
flig
lf_fli
gfr_
flab
lf_fla
b fpi
fpg
Parameter
0.00
0.05
0.10
0.15
0.20
0.25
S iGSA comparison for PCE and Tree-LSTM RNN
PCEMLPLSTM RNNTree LSTM RNN
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 26 / 28
Tree RNN more accurate than PCE and traditional MLmethods
Computed Mean RMS for each Surrogate
Method Train (Daily/Month) Val (Daily/Month)
PCE (N/A)/35% (N/A)/46%MLP 19/14% 32/20%
LSTM 14/10% 21/16%Tree-LSTM 6/2% 9/5%
Tree-LSTM outperforms PCE, MLP and LSTM-RNN
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 27 / 28
Conclusions and Future Work Thank you!
Conclusions
Physics-based NN architecture outperforms traditional NN methodsand PCE
Using Tree-Structure LSTM gate can train on the noisy daily data
Qualitatively similar sensitivity results compared to PCE
Future Work
Employ the surrogate for calibration and optimal experimental design
Analyze multiple sites
Spatio-temporal physics-based RNN
Incorporate low-rank tensor structure into RNN
Vish. Ratnaswamy (Sandia National Labs) Tree-LSTM NN Surrogate Models 28 / 28
ELM Simulation Details
50°N
40°N
30°N
50°N
40°N
30°N
110°W 80°W
(i, j) (i+ 1, j) (i+ 2, j)
(i, j + 1)
(i+ 1, j + 1)
(i+ 2, j + 1)
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
ACMf(TM , Tm,
BTRAN,FSDS)
AutotrophicRespirationf(TM , Tm)
AllocationPhenologyf(TM , Tm)
LitterProcessesf(TM , Tm)
SOMDecomposition
f(TM , Tm)
nue (grass,tree)slatop (everg, decid)
fpgleaf C:N
leaf C:Nfroot C:N
livewd C:Nbr mr
q10 mrrg frac
cstor tau
froot leafstem leaf
croot stemf livewd
gdd critcrit daylndays onndays offleaf long
froot longlwtop annfstor2tran
r mortk l(1,2,3)
k fragflig(cwd,fr,lf)
flab(lf,fr)fpi
q10 hr
q10 hrk som(1,2,3,4)
rf(l1s1,l2s2,l3s3)rf(s1s2,s2s3,s3s4)
soil4ci
GPP Rg, Rm
NPP
Litter1-3
SOM1-4
HR
Leaf(LAI)
stem
root
NEE
drivers cell (i+ 1, j + 1)
dayk−1 dayk dayk+1
drivers cell (i, j + 1)
dayk−1 dayk dayk+1
drivers cell (i+ 2, j + 1)
dayk−1 dayk dayk+1
drivers cell (i+ 1, j)
dayk−1 dayk dayk+1
drivers cell (i, j)
dayk−1 dayk dayk+1
drivers cell (i+ 2, j)
dayk−1 dayk dayk+1
Global Sensitivity Analysis
Y = f (X1,X2,X3, ...,XN)
Total Variance decomposition (normalized)
Var(Y ) = E [Var(Y |X )] + Var(E [Y |X ])
If Xi are independent: V (Y ) =N∑i=1
Vi +∑
1≤i≤j≤NVij + ..+ V1,..,p
Use Sobol indices → QOI’s variance to be decomposed based onvariances of inputs
Sobol Indices
Si =VarXi
(E (Y |Xi ))
Var(Y )First Sobol Indices
Sij =Varij
Var(Y )Second Order Sobol Indices
Method
PCE allows for analytical computationof Si
ML needs Monte Carlo Integration tocompute Si
Surrogate Modeling:Tree LSTM
LSTM Equations
fjk = σ(Wf xj +∑
Uklhjl)
ij = σ(Wixj +∑
Ulhjl)
oj = σ(Woxj + Ulhjl)
uj = tanh(Wcxj + Ulhjl)
cj =∑
fjl � cjl + ij � uj
hj = σj � tanh(cj)
(1)