An official website of the United States government
Emerging lithium-ion battery systems require high-fidelity electrochemical models for advanced control, diagnostics, and design. Accordingly, battery parameter estimation is an active research domain where novel algorithms are being developed to calibrate complex models from input-output data. Amidst these efforts, little focus has been placed on the fundamental mechanisms governing estimation accuracy, spurring the question, why is an estimate accurate or inaccurate? In response, we derive a generalized estimation error equation under the commonly adopted least-squares objective function, which reveals that the error can be represented as a combination of system uncertainties (i.e., in model, measurement, and parameter) and uncertainty-propagating sensitivity structures in the data. We then relate the error equation to conventional error analysis criteria, such as the Fisher information matrix, Cramér-Rao bound, and parameter sensitivity, to assess the benefits and limitations of each. The error equation is validated through several uni- and bivariate estimations of lithium-ion battery electrochemical parameters using experimental data. These results are also analyzed with the error equation to study the error compositions and parameter identifiability under different data. Finally, we show that adding target parameters to the estimation without increasing the amount of data intrinsically reduces the robustness of the results to system uncertainties.
more »
« less
This content will become publicly available on January 30, 2026
Algebraic identifiability of partial differential equation models
Differential equation models are crucial to scientific processes across many disciplines, and the values of model parameters are important for analyzing the behaviour of solutions. Identifying these values is known as a parameter estimation, a type of inverse problem, which has applications in areas that include industry, finance and biomedicine. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. Checking the global identifiability of model parameters is a useful tool when exploring the well-posedness of a given model. This problem has been intensively studied for ordinary differential equation models, where theory, several efficient algorithms and software packages have been developed. A comprehensive theory for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.
more »
« less
- PAR ID:
- 10618005
- Publisher / Repository:
- IOP Science
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 2
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 025022
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We extend the sensitivity rank condition (SERC), which tests for identifiability of smooth input-output systems, to a broader class of systems. Particularly, we build on our recently developed lexicographic SERC (L-SERC) theory and methods to achieve an identifiability test for differential-algebraic equation (DAE) systems for the first time, including nonsmooth systems. Additionally, we develop a method to determine the identifiable and non-identifiable parameter sets. We show how this new theory can be used to establish a (non-local) parameter reduction procedure and we show how parameter estimation problems can be solved. We apply the new methods to problems in wind turbine power systems and glucose-insulin kinetics.more » « less
-
We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters—hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics.more » « less
-
We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like otherwalk on spheres (WoS)algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces,etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters---hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics.more » « less
-
This paper investigates particle deposition driven by fluid evaporation in a single pore channel representative of those found in porous membranes. A moving boundary problem for the 2D heat equation is coupled with an evolution equation for the pore radius, and describes the physical processes of fluid evaporation, diffusion of the particle concentration, and deposition on the pore channel wall. Furthermore, a stochastic differential equation (SDE) approach based on a Brownian motion particle-level description of diffusion is used as a similar phenomenological representation to the partial differential equation (PDE) model. Sensitivity analysis reveals trends in dominant model parameters such as evaporation rate, deposition rate, the volume scaling coefficient, and investigates the monotonicity of concentration. Evaluations of the asymptotically reduced model and the SDE model against the 2D PDE model are done in terms of the pore radius and solute concentration over time. For further exploration, we apply the model to a 2D droplet as well with both deterministic and stochastic approaches.more » « less
