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Mantle convection plays a fundamental role in the long-term thermal evolution of terrestrial planets like Earth, Mars, Mercury and Venus. Yet, key parameters and initial conditions of the partial differential equations governing mantle convection are poorly constrained. This often requires a large sampling of the parameter space to determine which combinations can satisfy certain observational constraints. Traditionally, 1D models based on scaling laws used to parameterize convective heat transfer, have been used to tackle the computational bottleneck of high-fidelity forward runs in 2D or 3D. However, these are limited in the amount of physics they can model (e.g. depth dependent material properties are difficult to incorporate into these models) and predict only mean quantities such as the mean mantle temperature. In the first study, feed-forward neural networks (FNN) are trained on a large number of 2D simulations of a Mars-like planet to overcome these limitations. Given five key parameters governing mantle convection, the FNNs can reliably predict the evolution of the entire 1D laterally-averaged temperature profile in time. The five parameters that are varied throughout the thesis are: reference viscosity (which controls the overall vigor of convection), activation energy and activation volume of the diffusion creep rheology (which accounts for the pressure- and temperature-dependence of the viscosity, respectively), an enrichment factor for radiogenic elements in the crust (which controls the partitioning of the radiogenic elements in the mantle and the crust), and the initial radial distribution of the mantle temperature. In a related study, machine learning is used for probabilistic inversion. Using Mixture Density Networks (MDN), various sets of synthetic present-day observables for a Mars-like planet are inverted to infer the same five mantle convection parameters. It is shown that the constraints on a parameter can be quantified using the log-likelihood value, the negative of which is used as the loss function to train an MDN. The crustal enrichment factor of radiogenic heat sources can be constrained the best, followed by reference viscosity, when all the observables are available: core-mantle-boundary heat flux, surface heat flux, radial contraction, melt produced and duration of volcanism. The initial mantle temperature can be constrained if the radial contraction is available with at least some parts of the temperature profile. Activation energy of diffusion creep can only be weakly constrained, while the activation volume of diffusion creep cannot be constrained at all in the present setup. Different levels of uncertainty were also emulated in the observables and it was found that constraints on different parameters loosen with varying rates, with initial temperature being the most sensitive. The marginal MDN is modified to obtain a joint probability model, which captures the cross-correlations among all parameters. Finally surrogate modeling is revisited, but for predicting the full 2D temperature field, which contains more information in the form of convection structures such as rising hot plumes and sinking cold downwellings. Deep learning techniques are able to produce reliable parameterized surrogates (i.e. surrogates that predict state variables such as temperature based only on input parameters) of the solution of the underlying partial differential equations. First, convolutional autoencoders are used to compress the size of each temperature field and retain only the most important features in form of a latent space. Then, FNNs and long-short term memory networks (LSTM) are used to predict the compressed fields from the five mantle convection parameters. Proper orthogonal decomposition of the LSTM and FNN predictions shows that despite a lower mean relative accuracy, LSTMs capture the flow dynamics better than FNNs., Die Mantelkonvektion spielt eine grundlegende Rolle in der langfristigen thermischen Entwicklung von terrestrischen Planeten wie Erde, Mars, Merkur und Venus. Es ist jedoch schwer, die Schlüsselparameter und Anfangsbedingungen der partiellen Differentialgleichungen, die die Mantelkonvektion steuern, einzuschränken. Dies erfordert häufig eine große Stichprobe des Parameterraums, um zu bestimmen, welche Konvektionsparameter mit den Beobachtungen übereinstimmen. Traditionell wurden 1D-Modelle verwendet, um den rechnerischen Aufwand von High-Fidelity-Vorwärtsläufen in 2D oder 3D zu erleichtern. Diese basieren auf Skalierungsgesetzen, die den konvektiven Wärmetransport parametrisieren. Solche 1D-Modelle können aber nur eine begrenzte Menge an physikalischen Prozessen modellieren (z. B. lassen sich tiefenabhängige Materialeigenschaften nur schwer in diese Modelle integrieren) und nur durchschnittliche Ergebnisse wie der Mittelwert der Manteltemperatur vorhersagen. In der ersten Studie werden Feed-Forward Neural Networks (FNN) mit einer großen Anzahl von 2D-Simulationen eines marsähnlichen Planeten trainiert, um diese Einschränkungen zu überwinden. Angesichts von fünf Schlüsselparametern, die die Mantelkonvektion bestimmen, können die FNNs zuverlässig die zeitliche Entwicklung des gesamten seitlich gemittelten 1D-Temperaturprofils vorhersagen. Die fünf Parameter, die während der gesamten Arbeit variiert werden, sind: Referenzviskosität (die die Gesamtstärke der Konvektion steuert), Aktivierungsenergie und Aktivierungsvolumen der Diffusionskriechrheologie (die jeweils die Druck- und Temperaturabhängigkeit der Viskosität berücksichtigen), ein Anreicherungsfaktor für radiogene Elemente in der Kruste (der die Verteilung der radiogenen Elemente im Mantel und in der Kruste steuert) und die anfängliche radiale Verteilung der Manteltemperatur. In einer verwandten Studie wird maschinelles Lernen zur probabilistischen Inversion von Beobachtungen verwendet, um die Mantelkonvektionsparameter eines marsähnlichen Planeten einzuschränken. Mithilfe von Mixture Density Networks (MDN) werden verschiedene Datensätze heutiger synthetischer Observablen für einen marsähnlichen Planeten invertiert, um auf dieselben fünf Parameter zu schließen. Es wird gezeigt, dass die Einschränkungen eines Parameters unter Verwendung des Log-Likelihood-Werts quantifiziert werden können. Der Negativwert des Log-Likelihoods wird als Verlustfunktion zum Trainieren eines MDN verwendet. Der Krustenanreicherungsfaktor kann am besten bestimmt werden, gefolgt von der Referenzviskosität, wenn alle Observablen verfügbar sind: Kern-Mantel-Grenzwärmefluss, Oberflächenwärmefluss, radiale Kontraktion, produzierte Schmelze und Dauer des Vulkanismus. Die anfängliche Manteltemperatur kann bestimmt werden, wenn die radiale Kontraktion und zumindest einige Teile des Temperaturprofils verfügbar sind. Die Aktivierungsenergie des Diffusionskriechens kann nur schwach eingeschränkt werden, während das Aktivierungsvolumen des Diffusionskriechens in der vorliegenden Studie überhaupt nicht bestimmt werden kann. In den Beobachtungen wurden auch unterschiedliche Unsicherheitsgrade emuliert, und es wurde festgestellt, dass sich die Einschränkungen für verschiedene Parameter unterschiedlich schnell lockern, wobei die Anfangstemperatur am empfindlichsten ist. Die marginale MDN wird modifiziert, um ein gemeinsames Wahrscheinlichkeitsmodell zu erhalten, das die Kreuzkorrelationen zwischen allen Parametern erfasst. Schließlich wird die Surrogatemodellierung erneut aufgegriffen, jedoch für die Vorhersage des vollständigen 2D-Temperaturfelds, das mehr Informationen in Form von Konvektionsstrukturen wie heißen Schwaden und kalten Abwärtsströmungen enthält. Deep-Learning-Algorithmen sollen in der Lage sein, zuverlässige parametrisierte Surrogate (d. h. Surrogate, die Zustandsvariablen wie Temperatur nur auf der Grundlage von Parametern vorhersagen) zu erzeugen. Zunächst werden Convolutional-Autoencoder verwendet, um die Größe jedes Temperaturfelds zu komprimieren. Dann werden FNNs und Long-Short-Term Memory Netze (LSTM) verwendet, um die komprimierten Felder aus den fünf Mantelkonvektionsparametern vorherzusagen. Die Proper Orthogonal Decomposition der LSTM- und FNN-Vorhersagen zeigt, dass LSTMs trotz einer geringeren durchschnittlichen relativen Genauigkeit die Strömungsdynamik besser erfassen als FNNs.

Deep Learning has become overly complicated and has enjoyed stellar success in solving several classical problems like image classification, object detection, etc. Several methods for explaining these decisions have been proposed. Black-box methods to generate saliency maps are particularly interesting due to the fact that they do not utilize the internals of the model to explain the decision. Most black-box methods perturb the input and observe the changes in the output. We formulate saliency map generation as a sequential search problem and leverage upon Reinforcement Learning (RL) to accumulate evidence from input images that most strongly support decisions made by a classifier. Such a strategy encourages to search intelligently for the perturbations that will lead to high-quality explanations. While successful black box explanation approaches need to rely on heavy computations and suffer from small sample approximation, the deterministic policy learned by our method makes it a lot more efficient during the inference. Experiments on three benchmark datasets demonstrate the superiority of the proposed approach in inference time over state-of-the-arts without hurting the performance. Project Page: https://cvir.github.io/projects/rexl.html, Accepted in NeurIPS 2021 workshop on eXplainable AI approaches for debugging and diagnosis. Project Page: https://cvir.github.io/projects/rexl.html

Knowledge graphs (KGs) have helped neural models improve performance on various knowledge-intensive tasks, like question answering and item recommendation. By using attention over the KG, such KG-augmented models can also "explain" which KG information was most relevant for making a given prediction. In this paper, we question whether these models are really behaving as we expect. We show that, through a reinforcement learning policy (or even simple heuristics), one can produce deceptively perturbed KGs, which maintain the downstream performance of the original KG while significantly deviating from the original KG's semantics and structure. Our findings raise doubts about KG-augmented models' ability to reason about KG information and give sensible explanations., 13 pages, 11 figures

Thermal evolution simulations of planetary mantles in 2D and 3D are computationally intensive. A low-fidelity alternative is to use scaling laws based on boundary-layer theory to express Nusselt Number (Nu) as a function of Rayleigh Number (Ra). Such a Ra-Nu relation can be used to run `0D' parametrized evolution models by solving a simple energy balance equation. Yet scaling relations are available only for simple flows that cannot capture the full complexity of mantle dynamics. We propose leveraging Machine Learning to find a higher-dimensional mapping from five different parameters to the entire 1D temperature profile. The parameters are Ra, internal heating Ra, dissipation number and the maximum viscosity contrast between top and bottom due to temperature and pressure. We train a Neural Network (NN) to take these inputs and predict the resulting steady-state temperature profile. The training data comes from a subset of 20,000 compressible simulations on a 2D cylindrical grid. This results in predictions with an average error of 1.6% on the test set. The NN can potentially be used to build a 1D evolution model by stacking several steady-state temperature profiles together, with each prediction serving as an input at the next time-step.

This thesis extended the order of the reconstruction of state for convective fluxes used by Finite Volume (FV) algorithm in DLR’s next-generation CFD solver: Flucs, from constant and linear to quadratic and cubic. Two approaches for calculating derivatives were implemented in Flucs and some test cases were tried. To allow for integration of moments within each cell and a higher-order integration of fluxes, the mesh used by Discontinuous Galerkin (DG) was fed to the FV algorithm. Insufficient geometric treatment of the boundary cells and the dummy cells in FV is believed to be detrimental to the order of error reduction in NACA0012 case and the smooth bump case. In the smooth bump case, the FV algorithms failed to show higher than second order error reduction because of this reason. The order of the schemes away from the boundaries was verified with the Ehrenfried Vortex test case. For at least structured meshes and unstructured meshes with quads, schemes of order k approached k + 1 order accuracy on sufficiently fine meshes. The original goal of this thesis was partly accomplished and some further work in the code is expected. Detta examensarbete vidareutvecklade ordningen av rekonstruktion av tillståndet för konvektiva flöden som används av FV-algoritmen i DLR’s nästa generations CFD-lösare: Flucs, från konstant och linjär till kvadratisk och kubisk. Två metoder för beräkning av derivatorna genomfördes i Flucs och några testfall försökades. För att möjliggöra integrationen av momenten inom varje cell och integrationen av flöden på en högre ordning, matades det nät som används av DG till FV-algoritmen. Otillräcklig geometrisk behandling av gränscellerna och dummycellerna antas vara skadlig för ordning av felreduceringen i NACA0012-fallet och det Smmoth Bump-fallet. I det Smooth Bump-Fallet fick FV-algoritmerna inte visa högre än andra ordning av felreducering på grund av denna anledning. Ordningen av schemana bort från gränserna verifierades med Ehrenfried Vortex-testfallet. För åtminstone strukturerade nät och ostrukturerade nät med fyrkantiga celler nådde algorithmer med order k, k + 1 ordernoggrannhet på tillräckliga bra nät. Det ursprungliga målet för denna avhandling uppnåddes delvis och ytterligare arbete i koden förväntas.

The parameters and initial conditions governing mantle convection in terrestrial planets like Mars are poorly known meaning that one often needs to randomly vary several parameters to test which ones satisfy observational constraints. However, running forward models in 2D or 3D is computationally intensive to the point that it might prohibit a thorough scan of the entire parameter space. We propose using Machine Learning to find a low-dimensional mapping from input parameters to outputs. We use about 10,000 thermal evolution simulations with Mars-like parameters run on a 2D quarter cylindrical grid to train a fully-connected Neural Network (NN). We use the code GAIA (Hüttig et al., 2013) to solve the conservation equations of mantle convection for a fluid with Newtonian rheology and infinite Prandtl number under the Extended Boussinesq Approximation. The viscosity is calculated according to the Arrhenius law of diffusion creep (Hirth & Kohlstedt, 2003). The model also considers the effects of partial melting on the energy balance, including mantle depletion of heat producing-elements (Padovan et., 2017), as well as major phase transitions in the olivine system. To generate the dataset, we randomly vary 5 different parameters with respect to each other: thermal Rayleigh number, internal heating Rayleigh number, activation energy, activation volume and a depletion factor for heat-producing elements in the mantle. In order to train in time, we take the simplest possible approach, i.e., we treat time as another variable in our input vector. 80% of the dataset is used to train our NN, 10% is used to test different architectures and to avoid over-fitting, and the remaining 10% is used as test set to evaluate the error of the predictions. For given values of the five parameters, our NN can predict the resulting horizontally-averaged temperature profile at any time in the evolution, spanning 4.5 Ga with an average error under 0.3% on the test set. Tests indicate that with as few as 5% of the training samples (= simulations x time steps), one can achieve a test-error below 0.5%, suggesting that for this setup, one can potentially learn the mapping from fewer simulations. Finally, we ran a fourth batch of GAIA simulations and compared them to the output of our NN. In almost all cases, the instantaneous predictions of the 1D temperature profiles from the NN match those of the computationally expensive simulations extremely well, with an error below 0.5%.