geometry on Dan MacKinlay
https://danmackinlay.name/tags/geometry.html
Recent content in geometry on Dan MacKinlayHugo -- gohugo.ioen-usTue, 13 Apr 2021 16:24:05 +0800Machine learning for partial differential equations
https://danmackinlay.name/notebook/ml_pde.html
Tue, 13 Apr 2021 16:24:05 +0800https://danmackinlay.name/notebook/ml_pde.htmlLearning a PDE Deterministic PINN Stochastic PINN Weak formulation Learning a PDE forward operator Fourier neural operator DeepONet Advection-diffusion PDEs in particular Boundary conditions Inverse problems Differentiable solvers DeepXDE ADCME TenFEM JuliaFEM Trixi FEniCS taichi References Using statistical or machine learning approaches to solve PDEs, and maybe even to perform inference through them. There are various approaches here, which I will document on an ad hoc basis as I need them.Hydrology models
https://danmackinlay.name/notebook/hydrology.html
Tue, 13 Apr 2021 14:36:12 +0800https://danmackinlay.name/notebook/hydrology.htmlTools Simulation Geostats Framework Datasets References Refugees from Brazil’s Grande Seca drought of 1878
TBD: Ground water hydrology, surface water hydrology, coastal water hydrology… UCSB Climate hazards data.
Tools Simulation AFAICT the go-to applied reference here is Anderson, Woessner, and Hunt (2015).
MODFLOW
MODFLOW is the USGS’s modular hydrologic model. MODFLOW is considered an international standard for simulating and predicting groundwater conditions and groundwater/surface-water interactions.Machine learning for physical sciences
https://danmackinlay.name/notebook/ml_physics.html
Fri, 09 Apr 2021 11:48:30 +0800https://danmackinlay.name/notebook/ml_physics.htmlML for PDEs Causality, identifiability, and observational data Likelihood free inference Emulation approaches The other direction: What does physics say about learning? But statistics is ML Applications References Consider a spherical flame
In physics, typically, we are concerned with identifying True Parameters for Universal Laws, applicable without prejudice across all the cosmos. We are hunting something like the Platonic ideals that our experiments are poor shadows of.Convolutional Gaussian processes
https://danmackinlay.name/notebook/gp_convolution.html
Mon, 01 Mar 2021 17:08:51 +1100https://danmackinlay.name/notebook/gp_convolution.htmlConvolutions with respect to a non-stationary driving noise Varying convolutions with respect to a stationary white noise References Gaussian processes by convolution of noise with smoothing kernels, which is a kind of dual to defining them through covariances.
This is especially interesting because it can be made computationally convenient (we can enforce locality) and non-stationarity.
Convolutions with respect to a non-stationary driving noise H. K.Convolutional stochastic processes
https://danmackinlay.name/notebook/stochastic_convolution.html
Mon, 01 Mar 2021 16:13:24 +1100https://danmackinlay.name/notebook/stochastic_convolution.htmlReferences Stochastic processes generated by convolution of white noise with smoothing kernels, which is not unlike kernel density estimation where the “data” is random.
For now, I am mostly interested in certain special cases Gaussian process convolutionss and subordinator convolutions.
patrick-kidger/Deep-Signature-Transforms: Code for "Deep Signature Transforms" patrick-kidger/signatory: Differentiable computations of the signature and logsignature transforms, on both CPU and GPU. References Bolin, David.Multi-output Gaussian process regression
https://danmackinlay.name/notebook/gp_regression_functional.html
Mon, 07 Dec 2020 20:43:06 +1100https://danmackinlay.name/notebook/gp_regression_functional.htmlReferences In which I discov Learning operators via GPs.
References Brault, Romain, Florence d’Alché-Buc, and Markus Heinonen. 2016. “Random Fourier Features for Operator-Valued Kernels.” In Proceedings of The 8th Asian Conference on Machine Learning, 110–25. http://arxiv.org/abs/1605.02536. Brault, Romain, Néhémy Lim, and Florence d’Alché-Buc. n.d. “Scaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.” Accessed August 31, 2016. https://aaltd16.irisa.fr/files/2016/08/AALTD16_paper_11.pdf. Brouard, Céline, Marie Szafranski, and Florence D’Alché-Buc.Bushfire models
https://danmackinlay.name/notebook/bushfires.html
Wed, 19 Aug 2020 11:34:20 +1000https://danmackinlay.name/notebook/bushfires.htmlReferences american forest fire 1904
Press: Silicon valley wildfire spotting.
References Hilton, J. E., A. L. Sullivan, W. Swedosh, J. Sharples, and C. Thomas. 2018. “Incorporating Convective Feedback in Wildfire Simulations Using Pyrogenic Potential.” Environmental Modelling & Software 107 (September): 12–24. https://doi.org/10.1016/j.envsoft.2018.05.009. Mandel, Jan, Jonathan D. Beezley, Janice L. Coen, and Minjeong Kim. 2009. “Data Assimilation for Wildland Fires.” IEEE Control Systems Magazine 29 (3): 47–65.Hamiltonian and Langevin Monte Carlo
https://danmackinlay.name/notebook/hamiltonian_monte_carlo.html
Thu, 12 Jul 2018 21:07:16 +1000https://danmackinlay.name/notebook/hamiltonian_monte_carlo.htmlLangevin Monte Carlo To file References Hamiltonians, energy conservation in sampling. Handy. Summary would be nice.
Michael Betancourt’s heuristic explanation of Hamiltonian Monte Carlo: sets of high mass, no good - we need the “typical set”, a set whose product of differential volume and density is high. Motivates Markov Chain Monte Carlo on this basis, a way of exploring typical set given points already in it, or getting closer to the typical set if starting without.Differential geometry, geometric algebra etc
https://danmackinlay.name/notebook/differential_geometry.html
Thu, 13 Oct 2016 10:10:23 +1100https://danmackinlay.name/notebook/differential_geometry.htmlDifferential forms formalism Clifford Algebra formalism References 🏗 A pragmatic guide to manifold wrangling, whatever your formalism.
Key words: Clifford algebra, differential forms… Bollu’s Handy list of differential geometry definitions.
See information geometry.
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Differential forms formalism Flanders, Differential forms (Flanders 1989) is a wildly popular classic available for cheap used Frankel, the geometry of physics (Frankel 2011) attempts to update Flanders, trading slightly improved pedagogy for slightly more physics focus.Curved exponential families
https://danmackinlay.name/notebook/curved_exponential_families.html
Tue, 19 Apr 2016 21:43:04 +0800https://danmackinlay.name/notebook/curved_exponential_families.htmlReferences Curved exponential families generalise exponential families while preserving some of the magic, I guess?
Credited to (Efron 2004) and used in Efron (1978) to show some nice behaviour for degrees of freedom-based penalties. Will read soon.
Examples?
References Efron, Bradley. 1975. “Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency).” The Annals of Statistics 3 (6): 1189–1242. https://doi.org/10.1214/aos/1176343282. ———. 1978.Geometry of fitness landscapes
https://danmackinlay.name/notebook/geometry_of_fitness_landscapes.html
Wed, 17 Jun 2015 12:56:16 +0200https://danmackinlay.name/notebook/geometry_of_fitness_landscapes.htmlSee also knowledge topology, configuration space of the economy, evolution.
What “shape” is the fitness landscape explored by agents in an evolutionary process? In simple optimisation problems without interaction? In multi-agent systems with interactions between agents? (i.e. with niche construction) In actually existing nature, in all its chaotic glory?
Typically, genetic algorithms are implemented fixed-length genomes floating values in the range \([0,1]\) — the landscape is \([0,1]^n\) for genome of length \(n\).Configuration space of the economy
https://danmackinlay.name/notebook/configuration_space_of_the_economy.html
Wed, 07 Sep 2011 06:43:11 +0000https://danmackinlay.name/notebook/configuration_space_of_the_economy.htmlSee also:
innovation (I should merge these.) material basis of the economy Let’s keep our problem simple and bound our considerations at the earth’s atmosphere, considering solar radiation as exogenous. Now, this is not an undifferentiated mass of disordered particles, but rather a hodge-podge of both order and disorder. At some points it is highly simple and at other highly complicated. In the midst of this system, a part of the system, is the human economy, and the behaviour of that system is of great interest to me.