machine learning meets data assimilation...•data assimilation has firm basis in bayes theorem....
TRANSCRIPT
Data Assimilation for NWPProblem is how to provide best estimate and uncertainty estimates of physically related high-dimensional fields, e.g. global NWP, or high-resolution regional NWP.
.
This is different from ‘down-scaling’ in which only one or a few variables are to be estimated, typically from model forecast andobservations.
Photo by Brian Cook on Unsplash
Data Assimilation for NWP
Model forecastHigh-dimensionalUncertaintyPhysical constraints
ObservationsHigh-dimensionalUncertainty
Data AssimilationSolid maths (Bayes Theorem)Complex (slow)
AnalysisHigh-dimensionalUncertainty
Deep Learning for NWP?
Training dataHuge dimensional(e.g. model forecasts, observations)
Real dataHigh-dimensional
e.g Convolutional NNNot well understoodNo extrapolationFast
Analysis/forecastLow-dimensionalNo uncertainty?
Deep learning: Uncertainty quantification
Deep learning provides a nonlinear map from input to output :
DA is not a discrete problem, so uncertainty quantification is relatively easy.The noise term can be estimated from the test data (DL).
UQ for the output y can be determined from known uncertainty in the input x via sampling and propagation through the network:
We know how to do this… !
y = f(x) + ⌘<latexit sha1_base64="dWD5WZh7RMF/uT4X9zlVwWP1sCk=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEIFaEkVdCLUPTisYL9gDaUzXbTLt1swu5GrKG/xIsHRbz6U7z5b9y2OWjrg4HHezPMzPNjzpR2nG8rt7K6tr6R3yxsbe/sFu29/aaKEklog0Q8km0fK8qZoA3NNKftWFIc+py2/NHN1G89UKlYJO71OKZeiAeCBYxgbaSeXRyjKxSUH0/QKepSjXt2yak4M6Bl4makBBnqPfur249IElKhCcdKdVwn1l6KpWaE00mhmygaYzLCA9oxVOCQKi+dHT5Bx0bpoyCSpoRGM/X3RIpDpcahbzpDrIdq0ZuK/3mdRAeXXspEnGgqyHxRkHCkIzRNAfWZpETzsSGYSGZuRWSIJSbaZFUwIbiLLy+TZrXinlWqd+el2nUWRx4O4QjK4MIF1OAW6tAAAgk8wyu8WU/Wi/Vufcxbc1Y2cwB/YH3+ABPPkWs=</latexit>
⌘<latexit sha1_base64="jpVdjwnRsHnjvYQzwTGHXHUPFLk=">AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/YA2lM120y7d3YTdiVBC/4IXD4p49Q9589+YtDlo64OBx3szzMwLYiksuu63s7a+sbm1Xdop7+7tHxxWjo7bNkoM4y0Wych0A2q5FJq3UKDk3dhwqgLJO8HkLvc7T9xYEelHnMbcV3SkRSgYxVzqc6SDStWtuXOQVeIVpAoFmoPKV38YsURxjUxSa3ueG6OfUoOCST4r9xPLY8omdMR7GdVUceun81tn5DxThiSMTFYayVz9PZFSZe1UBVmnoji2y14u/uf1Egxv/FToOEGu2WJRmEiCEckfJ0NhOEM5zQhlRmS3EjamhjLM4ilnIXjLL6+Sdr3mXdbqD1fVxm0RRwlO4QwuwINraMA9NKEFDMbwDK/w5ijnxXl3Phata04xcwJ/4Hz+AAowjjw=</latexit>
y + �y = f(x+ �x) + ⌘<latexit sha1_base64="/oZiCbZrXJd2hUwmbhc3trWuwck=">AAACDnicbZDJSgNBEIZ7XGPcoh69NIZARAgzUdCLEPTiMYJZIAmhp1OTNOlZ6K6RDCFP4MVX8eJBEa+evfk2dhZQE39o+Piriur63UgKjbb9ZS0tr6yurac20ptb2zu7mb39qg5jxaHCQxmquss0SBFABQVKqEcKmO9KqLn963G9dg9KizC4wySCls+6gfAEZ2isdiaX0BPa7IBERhN6Sb384McYHI8ZkLUzWbtgT0QXwZlBlsxUbmc+m52Qxz4EyCXTuuHYEbaGTKHgEkbpZqwhYrzPutAwGDAfdGs4OWdEc8bpUC9U5gVIJ+7viSHztU5813T6DHt6vjY2/6s1YvQuWkMRRDFCwKeLvFhSDOk4G9oRCjjKxADjSpi/Ut5jinE0CaZNCM78yYtQLRac00Lx9ixbuprFkSKH5IjkiUPOSYnckDKpEE4eyBN5Ia/Wo/VsvVnv09YlazZzQP7I+vgGR5qZGg==</latexit>
Deep learning: Extrapolation
Deep learning minimizes a costfunction
Good for input within space spanned by training data, bad for extrapolation.Bring in physical constraints between output variables:
Starts to look like Data Assimilation, e.g. 3DVar !
J(w) =X
i
1
ri(yNN
i � yobsi )2 + �wTw<latexit sha1_base64="IiSZplsBaBmYh2a9qr6abliV290=">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</latexit>
J(w) =X
i
1
ri(yNN
i � yobsi )2 + �wTw + µg(yNN )<latexit sha1_base64="h10Zd/VHEXr1fcgYSeP723MosFU=">AAACYHicbVFNS8MwGE47P+acOvWml+AQNsTRTkEvguhFPIwJmxusW0mzdAaTtiSpo5T+SW8evPhLTOfQ6Xwh8OR53q888SJGpbKsN8MsrKyurRc3Spvlre2dyu7eowxjgUkXhywUfQ9JwmhAuooqRvqRIIh7jPS859tc770QIWkYdFQSkSFHk4D6FCOlKbcyva85HKknz0+nWR1eQUfG3KXQ8QXCqZ2lwqVZLRmlrVam6VOoYehJjeujJjyBDtOzxgj+NBl1Fi55Bo/h5HtIks1a1d1K1WpYs4DLwJ6DKphH2628OuMQx5wECjMk5cC2IjVMkVAUM5KVnFiSCOFnNCEDDQPEiRymM4MyeKyZMfRDoU+g4IxdrEgRlzLhns7M95R/tZz8TxvEyr8cpjSIYkUC/DXIjxlUIczdhmMqCFYs0QBhQfWuED8hba3Sf1LSJth/n7wMHpsN+6zRfDivXt/M7SiCQ3AEasAGF+Aa3IE26AIM3o2CUTa2jA+zaO6Yu1+ppjGv2Qe/wjz4BNl9tT8=</latexit>
Deep learning: Extrapolation
Build physics-based forecasts into costfunction, e.g. Reservoir computing (ML) and physics-based model:
(Pathak et al., 2018)
Explore ML techniques within DA
Powerful ML techniques:• Kernel embeddings• Stochastic gradient descent • Deep learning• Etc ...!
Ensemble of model states Ensemble of model statesbefore observations after observations
Particle filter
Nonlinear Data Assimilation (Retrievals)
Particle flows
The particles, so full atmospheric states, are propagated in artificial time s via
The question is: How to choose the vector flow field ?Need an efficient algorithms that iteratively decrease the distance betweenthe pdf of the particles and the posterior pdf (solution to Bayes Theorem).
Use kernel embedding (ML) of the flow field:
fs(x)<latexit sha1_base64="z02aMt//0Wo0nwuhPxiGgLdfHOk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBHqpeyKoMeiF48V7Ae0S8mm2TY2myxJVixL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfaGV1bX1js7BV3N7Z3dsvHRw2jUo1ZQ2qhNLtkBgmuGQNy61g7UQzEoeCtcLRzdRvPTJtuJL3dpywICYDySNOiXVSM+qZytNZr1T2qt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmmeV32v6t9dlGvXeRwFOIYTqIAPl1CDW6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnD/gtjrc=</latexit><latexit sha1_base64="z02aMt//0Wo0nwuhPxiGgLdfHOk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBHqpeyKoMeiF48V7Ae0S8mm2TY2myxJVixL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfaGV1bX1js7BV3N7Z3dsvHRw2jUo1ZQ2qhNLtkBgmuGQNy61g7UQzEoeCtcLRzdRvPTJtuJL3dpywICYDySNOiXVSM+qZytNZr1T2qt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmmeV32v6t9dlGvXeRwFOIYTqIAPl1CDW6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnD/gtjrc=</latexit><latexit sha1_base64="z02aMt//0Wo0nwuhPxiGgLdfHOk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBHqpeyKoMeiF48V7Ae0S8mm2TY2myxJVixL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfaGV1bX1js7BV3N7Z3dsvHRw2jUo1ZQ2qhNLtkBgmuGQNy61g7UQzEoeCtcLRzdRvPTJtuJL3dpywICYDySNOiXVSM+qZytNZr1T2qt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmmeV32v6t9dlGvXeRwFOIYTqIAPl1CDW6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnD/gtjrc=</latexit><latexit sha1_base64="z02aMt//0Wo0nwuhPxiGgLdfHOk=">AAAB7XicbVBNSwMxEJ34WetX1aOXYBHqpeyKoMeiF48V7Ae0S8mm2TY2myxJVixL/4MXD4p49f9489+YtnvQ1gcDj/dmmJkXJoIb63nfaGV1bX1js7BV3N7Z3dsvHRw2jUo1ZQ2qhNLtkBgmuGQNy61g7UQzEoeCtcLRzdRvPTJtuJL3dpywICYDySNOiXVSM+qZytNZr1T2qt4MeJn4OSlDjnqv9NXtK5rGTFoqiDEd30tskBFtORVsUuymhiWEjsiAdRyVJGYmyGbXTvCpU/o4UtqVtHim/p7ISGzMOA5dZ0zs0Cx6U/E/r5Pa6CrIuExSyySdL4pSga3C09dxn2tGrRg7Qqjm7lZMh0QTal1ARReCv/jyMmmeV32v6t9dlGvXeRwFOIYTqIAPl1CDW6hDAyg8wDO8whtS6AW9o4956wrKZ47gD9DnD/gtjrc=</latexit>
dx
ds= fs(x)
<latexit sha1_base64="xnFnbKnNIQgCvl6KTIzrt9aU7ZE=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWoW5KIoJuhKIblxXsA9oQJpNJO3TyYGYiLSEbf8WNC0Xc+hnu/BunbRbaeuDC4Zx7ufceL+FMKsv6Nkorq2vrG+XNytb2zu6euX/QlnEqCG2RmMei62FJOYtoSzHFaTcRFIcepx1vdDv1O49USBZHD2qSUCfEg4gFjGClJdc86gcCk8wf55kvc3SNAlfWxmfINatW3ZoBLRO7IFUo0HTNr74fkzSkkSIcS9mzrUQ5GRaKEU7zSj+VNMFkhAe0p2mEQyqdbPZAjk614qMgFroihWbq74kMh1JOQk93hlgN5aI3Ff/zeqkKrpyMRUmqaETmi4KUIxWjaRrIZ4ISxSeaYCKYvhWRIdaJKJ1ZRYdgL768TNrndduq2/cX1cZNEUcZjuEEamDDJTTgDprQAgI5PMMrvBlPxovxbnzMW0tGMXMIf2B8/gCxppXK</latexit><latexit sha1_base64="xnFnbKnNIQgCvl6KTIzrt9aU7ZE=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWoW5KIoJuhKIblxXsA9oQJpNJO3TyYGYiLSEbf8WNC0Xc+hnu/BunbRbaeuDC4Zx7ufceL+FMKsv6Nkorq2vrG+XNytb2zu6euX/QlnEqCG2RmMei62FJOYtoSzHFaTcRFIcepx1vdDv1O49USBZHD2qSUCfEg4gFjGClJdc86gcCk8wf55kvc3SNAlfWxmfINatW3ZoBLRO7IFUo0HTNr74fkzSkkSIcS9mzrUQ5GRaKEU7zSj+VNMFkhAe0p2mEQyqdbPZAjk614qMgFroihWbq74kMh1JOQk93hlgN5aI3Ff/zeqkKrpyMRUmqaETmi4KUIxWjaRrIZ4ISxSeaYCKYvhWRIdaJKJ1ZRYdgL768TNrndduq2/cX1cZNEUcZjuEEamDDJTTgDprQAgI5PMMrvBlPxovxbnzMW0tGMXMIf2B8/gCxppXK</latexit><latexit sha1_base64="xnFnbKnNIQgCvl6KTIzrt9aU7ZE=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWoW5KIoJuhKIblxXsA9oQJpNJO3TyYGYiLSEbf8WNC0Xc+hnu/BunbRbaeuDC4Zx7ufceL+FMKsv6Nkorq2vrG+XNytb2zu6euX/QlnEqCG2RmMei62FJOYtoSzHFaTcRFIcepx1vdDv1O49USBZHD2qSUCfEg4gFjGClJdc86gcCk8wf55kvc3SNAlfWxmfINatW3ZoBLRO7IFUo0HTNr74fkzSkkSIcS9mzrUQ5GRaKEU7zSj+VNMFkhAe0p2mEQyqdbPZAjk614qMgFroihWbq74kMh1JOQk93hlgN5aI3Ff/zeqkKrpyMRUmqaETmi4KUIxWjaRrIZ4ISxSeaYCKYvhWRIdaJKJ1ZRYdgL768TNrndduq2/cX1cZNEUcZjuEEamDDJTTgDprQAgI5PMMrvBlPxovxbnzMW0tGMXMIf2B8/gCxppXK</latexit><latexit sha1_base64="xnFnbKnNIQgCvl6KTIzrt9aU7ZE=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWoW5KIoJuhKIblxXsA9oQJpNJO3TyYGYiLSEbf8WNC0Xc+hnu/BunbRbaeuDC4Zx7ufceL+FMKsv6Nkorq2vrG+XNytb2zu6euX/QlnEqCG2RmMei62FJOYtoSzHFaTcRFIcepx1vdDv1O49USBZHD2qSUCfEg4gFjGClJdc86gcCk8wf55kvc3SNAlfWxmfINatW3ZoBLRO7IFUo0HTNr74fkzSkkSIcS9mzrUQ5GRaKEU7zSj+VNMFkhAe0p2mEQyqdbPZAjk614qMgFroihWbq74kMh1JOQk93hlgN5aI3Ff/zeqkKrpyMRUmqaETmi4KUIxWjaRrIZ4ISxSeaYCKYvhWRIdaJKJ1ZRYdgL768TNrndduq2/cX1cZNEUcZjuEEamDDJTTgDprQAgI5PMMrvBlPxovxbnzMW0tGMXMIf2B8/gCxppXK</latexit>
fs(x) = hK(x, ·), fs(·)iF<latexit sha1_base64="za4jNYIBeSdV/jNrkrvuCo8PpIo=">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</latexit><latexit sha1_base64="za4jNYIBeSdV/jNrkrvuCo8PpIo=">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</latexit><latexit sha1_base64="za4jNYIBeSdV/jNrkrvuCo8PpIo=">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</latexit><latexit sha1_base64="za4jNYIBeSdV/jNrkrvuCo8PpIo=">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</latexit>
Particle flow in action
Results exploring ML in DA
True meridional velocity
Particle mean meridional velocity
Stochastic gradient descent (ML) in DA
We tried stochastic gradient descent methods like ADAM, but found difficulties with physical balances.Instead we used an approximate Newton method:
Convergence accelerated when we add momentum memory as in RMSProp. First calculate momentum update:
Then update move in state space as:
A�1�x = �rdist<latexit sha1_base64="e1w9Lkm0MtCD7Fs0EYlucCObMZ8=">AAACCHicbVC7SgNBFJ31GeNr1dLCwSDYJOxGQRshPgrLCOYB2TXcnUySIbOzy8ysGJaUNv6KjYUitn6CnX/j5FFo4oGBwzn3cuecIOZMacf5tubmFxaXljMr2dW19Y1Ne2u7qqJEElohEY9kPQBFORO0opnmtB5LCmHAaS3oXQ792j2VikXiVvdj6ofQEazNCGgjNe2987s07w6wd0W5BvyAz3AeewICDrhlzjftnFNwRsCzxJ2QHJqg3LS/vFZEkpAKTTgo1XCdWPspSM0Ip4OslygaA+lBhzYMFRBS5aejIAN8YJQWbkfSPKHxSP29kUKoVD8MzGQIuqumvaH4n9dIdPvUT5mIE00FGR9qJxzrCA9bMVElJZr3DQEimfkrJl2QQLTpLmtKcKcjz5JqseAeFYo3x7nSxaSODNpF++gQuegEldA1KqMKIugRPaNX9GY9WS/Wu/UxHp2zJjs76A+szx+aJ5fQ</latexit>
vn+1 = �vn � (1� �)Ardist<latexit sha1_base64="K4mOBim8h6R6kpbxAA+BolD1F+E=">AAACKHicbVBNSwMxFMz6WetX1aOXYBEUsexWQS9i1YvHCrYK3bW8TbM2NJtdkmyhLP05XvwrXkQU6dVfYrrtobYOBIaZ98ib8WPOlLbtgTU3v7C4tJxbya+urW9sFra26ypKJKE1EvFIPvqgKGeC1jTTnD7GkkLoc/rgd26G/kOXSsUica97MfVCeBYsYAS0kZqFSzcE3faDtNt/SsWR08cX2PWpBjxhCHyMD5zjTD/EV9gV4HPALXNes1C0S3YGPEucMSmiMarNwofbikgSUqEJB6Uajh1rLwWpGeG0n3cTRWMgHXimDUMFhFR5aRa0j/eN0sJBJM0TGmfq5EYKoVK90DeTw+vVtDcU//MaiQ7OvZSJONFUkNFHQcKxjvCwNRNVUqJ5zxAgkplbMWmDBKJNt3lTgjMdeZbUyyXnpFS+Oy1Wrsd15NAu2kMHyEFnqIJuURXVEEEv6A19oi/r1Xq3vq3BaHTOGu/soD+wfn4BWVqk4Q==</latexit>
�x = ⌘vn+1<latexit sha1_base64="NaGdUDDbq9Q2gVUMEUf5zdouO6c=">AAACCXicbZDLSsNAFIYnXmu9RV26GSyCIJSkCroRirpwWcFeoKllMj1ph04mYWZSLKFbN76KGxeKuPUN3Pk2TtoutPWHgY//nMOc8/sxZ0o7zre1sLi0vLKaW8uvb2xubds7uzUVJZJClUY8kg2fKOBMQFUzzaERSyChz6Hu96+yen0AUrFI3OlhDK2QdAULGCXaWG0be9fANcEP+AJ7YMALie75QToY3afi2B217YJTdMbC8+BOoYCmqrTtL68T0SQEoSknSjVdJ9atlEjNKIdR3ksUxIT2SReaBgUJQbXS8SUjfGicDg4iaZ7QeOz+nkhJqNQw9E1ntqearWXmf7VmooPzVspEnGgQdPJRkHCsI5zFgjtMAtV8aIBQycyumPaIJFSb8PImBHf25HmolYruSbF0e1ooX07jyKF9dICOkIvOUBndoAqqIooe0TN6RW/Wk/VivVsfk9YFazqzh/7I+vwBIzuZXA==</latexit>
Conclusions
• Data assimilation has firm basis in Bayes Theorem.
• Deep Learning can benefit strongly from DA techniques
• DA and ML should merge, within the Bayesian framework.
• ML techniques are extremely useful for nonlinear DA, e.g. kernel embedding,
(elements of) stochastic gradient descent.
• More work needed to merge more completely.
• High-resolution ocean and atmospheric DA exploring ML techniques is underway.
8th International Symposium on Data Assimilation
Canvas Stadium, Colorado State University
Fort Collins, Colorado, USA
8th – 12th June, 2020