3 research outputs found
Multiscale Neural Operators for Solving Time-Independent PDEs
Time-independent Partial Differential Equations (PDEs) on large meshes pose
significant challenges for data-driven neural PDE solvers. We introduce a novel
graph rewiring technique to tackle some of these challenges, such as
aggregating information across scales and on irregular meshes. Our proposed
approach bridges distant nodes, enhancing the global interaction capabilities
of GNNs. Our experiments on three datasets reveal that GNN-based methods set
new performance standards for time-independent PDEs on irregular meshes.
Finally, we show that our graph rewiring strategy boosts the performance of
baseline methods, achieving state-of-the-art results in one of the tasks.Comment: The Symbiosis of Deep Learning and Differential Equations III @
NeurIPS 202
Curve Your Enthusiasm: Concurvity Regularization in Differentiable Generalized Additive Models
Generalized Additive Models (GAMs) have recently experienced a resurgence in
popularity due to their interpretability, which arises from expressing the
target value as a sum of non-linear transformations of the features. Despite
the current enthusiasm for GAMs, their susceptibility to concurvity - i.e.,
(possibly non-linear) dependencies between the features - has hitherto been
largely overlooked. Here, we demonstrate how concurvity can severly impair the
interpretability of GAMs and propose a remedy: a conceptually simple, yet
effective regularizer which penalizes pairwise correlations of the non-linearly
transformed feature variables. This procedure is applicable to any
differentiable additive model, such as Neural Additive Models or NeuralProphet,
and enhances interpretability by eliminating ambiguities due to self-canceling
feature contributions. We validate the effectiveness of our regularizer in
experiments on synthetic as well as real-world datasets for time-series and
tabular data. Our experiments show that concurvity in GAMs can be reduced
without significantly compromising prediction quality, improving
interpretability and reducing variance in the feature importances
Enhancing Multi-Objective Optimization through Machine Learning-Supported Multiphysics Simulation
Multiphysics simulations that involve multiple coupled physical phenomena
quickly become computationally expensive. This imposes challenges for
practitioners aiming to find optimal configurations for these problems
satisfying multiple objectives, as optimization algorithms often require
querying the simulation many times. This paper presents a methodological
framework for training, self-optimizing, and self-organizing surrogate models
to approximate and speed up Multiphysics simulations. We generate two
real-world tabular datasets, which we make publicly available, and show that
surrogate models can be trained on relatively small amounts of data to
approximate the underlying simulations accurately. We conduct extensive
experiments combining four machine learning and deep learning algorithms with
two optimization algorithms and a comprehensive evaluation strategy. Finally,
we evaluate the performance of our combined training and optimization pipeline
by verifying the generated Pareto-optimal results using the ground truth
simulations. We also employ explainable AI techniques to analyse our surrogates
and conduct a preselection strategy to determine the most relevant features in
our real-world examples. This approach lets us understand the underlying
problem and identify critical partial dependencies