7 research outputs found
One-shot learning for solution operators of partial differential equations
Discovering governing equations of a physical system, represented by partial
differential equations (PDEs), from data is a central challenge in a variety of
areas of science and engineering. Current methods require either some prior
knowledge (e.g., candidate PDE terms) to discover the PDE form, or a large
dataset to learn a surrogate model of the PDE solution operator. Here, we
propose the first learning method that only needs one PDE solution, i.e.,
one-shot learning. We first decompose the entire computational domain into
small domains, where we learn a local solution operator, and then find the
coupled solution via a fixed-point iteration. We demonstrate the effectiveness
of our method on different PDEs, and our method exhibits a strong
generalization property
Physics-informed learning of governing equations from scarce data
Harnessing data to discover the underlying governing laws or equations that
describe the behavior of complex physical systems can significantly advance our
modeling, simulation and understanding of such systems in various science and
engineering disciplines. This work introduces a novel physics-informed deep
learning framework to discover governing partial differential equations (PDEs)
from scarce and noisy data for nonlinear spatiotemporal systems. In particular,
this approach seamlessly integrates the strengths of deep neural networks for
rich representation learning, physics embedding, automatic differentiation and
sparse regression to (1) approximate the solution of system variables, (2)
compute essential derivatives, as well as (3) identify the key derivative terms
and parameters that form the structure and explicit expression of the PDEs. The
efficacy and robustness of this method are demonstrated, both numerically and
experimentally, on discovering a variety of PDE systems with different levels
of data scarcity and noise accounting for different initial/boundary
conditions. The resulting computational framework shows the potential for
closed-form model discovery in practical applications where large and accurate
datasets are intractable to capture.Comment: 46 pages; 1 table, 6 figures and 3 extended data figures in main
text; 2 tables and 12 figures in supplementary informatio
Simulation Intelligence: Towards a New Generation of Scientific Methods
The original "Seven Motifs" set forth a roadmap of essential methods for the
field of scientific computing, where a motif is an algorithmic method that
captures a pattern of computation and data movement. We present the "Nine
Motifs of Simulation Intelligence", a roadmap for the development and
integration of the essential algorithms necessary for a merger of scientific
computing, scientific simulation, and artificial intelligence. We call this
merger simulation intelligence (SI), for short. We argue the motifs of
simulation intelligence are interconnected and interdependent, much like the
components within the layers of an operating system. Using this metaphor, we
explore the nature of each layer of the simulation intelligence operating
system stack (SI-stack) and the motifs therein: (1) Multi-physics and
multi-scale modeling; (2) Surrogate modeling and emulation; (3)
Simulation-based inference; (4) Causal modeling and inference; (5) Agent-based
modeling; (6) Probabilistic programming; (7) Differentiable programming; (8)
Open-ended optimization; (9) Machine programming. We believe coordinated
efforts between motifs offers immense opportunity to accelerate scientific
discovery, from solving inverse problems in synthetic biology and climate
science, to directing nuclear energy experiments and predicting emergent
behavior in socioeconomic settings. We elaborate on each layer of the SI-stack,
detailing the state-of-art methods, presenting examples to highlight challenges
and opportunities, and advocating for specific ways to advance the motifs and
the synergies from their combinations. Advancing and integrating these
technologies can enable a robust and efficient hypothesis-simulation-analysis
type of scientific method, which we introduce with several use-cases for
human-machine teaming and automated science
Automated Knowledge Discovery using Neural Networks
The natural world is known to consistently abide by scientific laws that can be expressed concisely in mathematical terms, including differential equations. To understand the patterns that define these scientific laws, it is necessary to discover and solve these mathematical problems after making observations and collecting data on natural phenomena. While artificial neural networks are powerful black-box tools for automating tasks related to intelligence, the solutions we seek are related to the concise and interpretable form of symbolic mathematics.
In this work, we focus on the idea of a symbolic function learner, or SFL. A symbolic function learner can be any algorithm that is able to produce a symbolic mathematical expression that aims to optimize a given objective function. By choosing different objective functions, the SFL can be tuned to handle different learning tasks. We present a model for an SFL that is based on neural networks and can be trained using deep learning. We then use this SFL to approach the computational task of automating discovery of scientific knowledge in three ways.
We first apply our symbolic function learner as a tool for symbolic regression, a curve-fitting problem that has traditionally been approached using genetic evolution algorithms. We show that our SFL performs competitively in comparison to genetic algorithms and neural network regressors on a sample collection of regression instances. We also reframe the problem of learning differential equations as a task in symbolic regression, and use our SFL to rediscover some equations from classical physics from data.
We next present a machine-learning based method for solving differential equations symbolically. When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions to solve differential equations while leveraging deep learning training methods. Unlike existing methods, our system does not require learning a language model over symbolic mathematics, making it scalable, compact, and easily adaptable for a variety of tasks and configurations. The system is designed to always return a valid symbolic formula, generating a useful approximation when an exact analytic solution to a differential equation is not or cannot be found. We demonstrate through examples the way our method can be applied on a number of differential equations that are rooted in the natural sciences, often obtaining symbolic approximations that are useful or insightful. Furthermore, we show how the system can be effortlessly generalized to find symbolic solutions to other mathematical tasks, including integration and functional equations.
We then introduce a novel method for discovering implicit relationships between variables in structured datasets in an unsupervised way. Rather than explicitly designating a causal relationship between input and output variables, our method finds mathematical relationships between variables without treating any variable as distinguished from any other. As a result, properties about the data itself can be discovered, rather than rules for predicting one variable from the others. We showcase examples of our method in the domain of geometry, demonstrating how we can re-discover famous geometric identities automatically from artificially generated data.
In total, this thesis aims to strengthen the connection between neural networks and problems in symbolic mathematics. Our proposed SFL is the main tool that we show can be applied to a variety of tasks, including but not limited to symbolic regression. We show how using this approach to symbolic function learning paves the way for future developments in automated scientific knowledge discovery