Abstract:
Waves are the most important physical process for information transfer. Traversing any kind of medium, they reveal minute details about their origin and path anywhere from the big bang to bats and big oil. While most wave processes obey relatively simple mathematical and often linear systems, accurate solutions in realistic 3D media rank amongst the hardest computational problems in science and engineering. This is mostly due to the exuberant scale of the problem, owing to multi-scale waves propagating over vast distances: Conventional wave problems can easily take up 10^15 degrees of freedom. It is thus paramount to devise strategies for exploiting computational efficiency, such as harnessing smoothness in material properties and the physical wavefield, as well as algorithmic and hardware acceleration. This talk will attempt to sketch a journey through orders of magnitude in scale and computational complexity, and present solutions tailored to each of these real-world problems, ending up with the most recent ideas on merging physics-informed neural networks with our knowledge of wave propagation. Seismology is a discpline quite uniquely positioned at the cusp between high-level computational problems and large, complex datasets of equal importance. To strike that balance and our challenge of robust inference, I won't resist the temptation to present some pathways in understanding savanna ecosystems with multimodal data, focussed on wild elephants.
Bio:
Tarje Nissen-Meyer is a Professor of Geophysics at the University of Oxford, Dept of Earth Sciences, and will be a Professor in Environmental Intelligence at the University of Exeter, Dept of Mathematics, starting October 2023. Originally trained as a physicist at the University of Munich and McGill University, Montreal, he obtained a PhD in computational seismology from Princeton University in 2008, devising a novel numerical method for wave propagation in axisymmetric media. Following postdocs at Caltech and Princeton, he joined ETH Zurich as a senior research scientist before coming to Oxford, and has since held visiting positions at Columbia and Stanford and is currently a Turing Fellow at London's Alan Turing Institute. With his group, he developed a range of numerical methods for wave propagation based on finite differences, spectral elements, Green's function databases, Lattice Boltzman and more recently using machine learning. He is a science team member on NASA's InSight mission, and has more recently focussed on research in the context of climate and biodiversity crises, with projects on wildlife monitoring in the African ecosystems (seismic tracking of elephants), monitoring of Antarctic iceshelves, and better understanding soil health for sustainable agriculture.
Summary
Focus: modeling wave behavior
Waves are an extremely common way for information to propagate over physical systems
Seismic
Wind
Acoustic
Elastic
Electromagnetic
Information propagation mechanisms
Waves: long-range
Touch: Mechanical
Hear: Acoustic
Sight: Electromagnetic
Others: short-range
Smell: Air Diffusion
Taste: Chemical dissolution
Linear wave systems
Special case of wave propagation
Wave amplitude is a linear function of the shock
This category of waves can be modeled more efficiently, making it possible to model them in complex, large-scale scenarios
Problem: 1015 degrees of freedom, for ~16 scenarios
Inference challenge:
How to compute posteriors for 106 parameters
Bayesian inference is typically limited to ~100
Application: Earth-system monitoring, observing/understanding
Interconnected complex systems with tipping points, noisy data, etc.
Goal: design adaptive solutions for global-scale problems
Data: quantity, quality, resolution, homogeneity
Processes:
Scales: Limited, gaps, space-time
Challenge: a complex system of interconnected complex systems
Can we justify any consistency properties that will make it possible to simplify the model?
Forward problem: well-defined event, full wave equation with unique solution, clear question to answer [focus of the talk]
Inverse problem: given observed wiggle infer where the wave comes from
Wave equation: multi-scale multi-physics PDE
Acceleration of displacement field = Divergence of Stress/Strain (assumed linear)
Linear form is good approximation far away from strong waving sources
Not good near earthquakes sources or where liquefaction occurs
Can add more terms to capture additional phenomena: gravity, earth rotation, deformations, etc.
Typical way to solve these: finite element approximation via a system of global ODEs in time
Data parsimony
Seismic waves: mostly bounce between core and surface around a sphere in a simple, symmetric way; little reflection within mantle
AxiSEM: http://seis.earth.ox.ac.uk/axisem/
Axisymmetric global wave propagation
3D solution of seismic wave propagation at the cost of a 2D solution by leveraging Earth spherical symmetry
Good approximation to a 3D solver (SpecFEM: https://specfem.org/)
3D solution is mostly symmetric but with local wiggles
Can be better approximated via a mostly 2D solution with a coarsely refined 3rd dimension (Fourier series expansion with few terms)
This makes it possible to
Efficiently simulate large earthquakes
Simulate 1Hz/10km scale waves at core-mantle boundary
Machine learning strategies for PDEs
Replace expensive forward/inverse solves fully or partially
Approximate multi-scale physics via dimensional reduction/surrogates
Reduce overall problem size
Explored entire space
PureML: Wavelet approximation
Pure PDE: AxiSEM
Hybrid:
Physics-informed Neural Networks (PINNs)
Train neural network using real PDE as a loss
Learned neural net is a good approximation of the PDE
But struggle in some conditions
FBPINN: Finite element analysis + PINNS
Finite-basis: domain decomposition, subdomain normalization, flexible training schedules
Sub-divides space into sub-networks
One network per type of wave propagation (e.g. core, mantle, surface)
Can decompose this into spatial regions or use other criteria
TerraPIN: Parsimony/AxiSEM3D + PINN
Regular neural nets in the radial direction
PINNs in the azimuthal direction
Real-world application:
1 million scenarios to get a probabilistics hazard model
Train Neural Network on dataset of source-receiver pairs
10 simulation runs of PDE model, with seismic 1 source each
9477 receivers per source
Output: amplitude spectra of full distribution of 1 million runs of neural model
End–to-end multiscale modeling
Using a wide range of planetary datasets to parameterize models
Induce realistic shaking and boundary conditions
Traditional approach: separate forward and inverse modeling steps
Use differentiable modeling frameworks (e.g. JAX) to combine them
Use back-propagation to tune structure, discretization and other attributes of the model