146 research outputs found
Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations
We propose a novel gray-box modeling algorithm for physical systems governed
by stochastic differential equations (SDE). The proposed approach, referred to
as the Deep Physics Corrector (DPC), blends approximate physics represented in
terms of SDE with deep neural network (DNN). The primary idea here is to
exploit DNN to model the missing physics. We hypothesize that combining
incomplete physics with data will make the model interpretable and allow better
generalization. The primary bottleneck associated with training surrogate
models for stochastic simulators is often associated with selecting the
suitable loss function. Among the different loss functions available in the
literature, we use the conditional maximum mean discrepancy (CMMD) loss
function in DPC because of its proven performance. Overall, physics-data fusion
and CMMD allow DPC to learn from sparse data. We illustrate the performance of
the proposed DPC on four benchmark examples from the literature. The results
obtained are highly accurate, indicating its possible application as a
surrogate model for stochastic simulators
A foundational neural operator that continuously learns without forgetting
Machine learning has witnessed substantial growth, leading to the development
of advanced artificial intelligence models crafted to address a wide range of
real-world challenges spanning various domains, such as computer vision,
natural language processing, and scientific computing. Nevertheless, the
creation of custom models for each new task remains a resource-intensive
undertaking, demanding considerable computational time and memory resources. In
this study, we introduce the concept of the Neural Combinatorial Wavelet Neural
Operator (NCWNO) as a foundational model for scientific computing. This model
is specifically designed to excel in learning from a diverse spectrum of
physics and continuously adapt to the solution operators associated with
parametric partial differential equations (PDEs). The NCWNO leverages a gated
structure that employs local wavelet experts to acquire shared features across
multiple physical systems, complemented by a memory-based ensembling approach
among these local wavelet experts. This combination enables rapid adaptation to
new challenges. The proposed foundational model offers two key advantages: (i)
it can simultaneously learn solution operators for multiple parametric PDEs,
and (ii) it can swiftly generalize to new parametric PDEs with minimal
fine-tuning. The proposed NCWNO is the first foundational operator learning
algorithm distinguished by its (i) robustness against catastrophic forgetting,
(ii) the maintenance of positive transfer for new parametric PDEs, and (iii)
the facilitation of knowledge transfer across dissimilar tasks. Through an
extensive set of benchmark examples, we demonstrate that the NCWNO can
outperform task-specific baseline operator learning frameworks with minimal
hyperparameter tuning at the prediction stage. We also show that with minimal
fine-tuning, the NCWNO performs accurate combinatorial learning of new
parametric PDEs
A Bayesian framework for discovering interpretable Lagrangian of dynamical systems from data
Learning and predicting the dynamics of physical systems requires a profound
understanding of the underlying physical laws. Recent works on learning
physical laws involve generalizing the equation discovery frameworks to the
discovery of Hamiltonian and Lagrangian of physical systems. While the existing
methods parameterize the Lagrangian using neural networks, we propose an
alternate framework for learning interpretable Lagrangian descriptions of
physical systems from limited data using the sparse Bayesian approach. Unlike
existing neural network-based approaches, the proposed approach (a) yields an
interpretable description of Lagrangian, (b) exploits Bayesian learning to
quantify the epistemic uncertainty due to limited data, (c) automates the
distillation of Hamiltonian from the learned Lagrangian using Legendre
transformation, and (d) provides ordinary (ODE) and partial differential
equation (PDE) based descriptions of the observed systems. Six different
examples involving both discrete and continuous system illustrates the efficacy
of the proposed approach
Multi-fidelity wavelet neural operator with application to uncertainty quantification
Operator learning frameworks, because of their ability to learn nonlinear
maps between two infinite dimensional functional spaces and utilization of
neural networks in doing so, have recently emerged as one of the more pertinent
areas in the field of applied machine learning. Although these frameworks are
extremely capable when it comes to modeling complex phenomena, they require an
extensive amount of data for successful training which is often not available
or is too expensive. However, this issue can be alleviated with the use of
multi-fidelity learning, where a model is trained by making use of a large
amount of inexpensive low-fidelity data along with a small amount of expensive
high-fidelity data. To this end, we develop a new framework based on the
wavelet neural operator which is capable of learning from a multi-fidelity
dataset. The developed model's excellent learning capabilities are demonstrated
by solving different problems which require effective correlation learning
between the two fidelities for surrogate construction. Furthermore, we also
assess the application of the developed framework for uncertainty
quantification. The results obtained from this work illustrate the excellent
performance of the proposed framework
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