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The optical filter is critical in many applications requiring wide-angle imaging perception. However, the transmission curve of the typical optical filter will change at an oblique incident angle due to the optical path of the incident light change. In this study, we propose a wide-angular tolerance optical filter design method based on the transfer matrix method and automatic differentiation. A novel optical merit function is proposed for simultaneous optimization at normal and oblique incidents. The simulation results demonstrate that such a wide-angular tolerance design can realize a similar transmittance curve at an oblique incident angle compared to a normal incident angle. Furthermore, how much improvement in a wide-angular optical filter design for oblique incident contributes to image segmentation remains unclear. Therefore, we evaluate several transmittance curves along with the U-Net structure for green pepper segmentation. Although our proposed method is not perfectly equal to the target design, it can achieve an average 50% smaller mean absolute error (MAE) than the original design at 20∘ oblique incident angle. In addition, the green pepper segmentation results show that wide-angular tolerance optical filter design improves the segmentation of the near-color object about 0.3% at 20∘ oblique incident angle compared to the previous design.

Differentiable Earth system models (ESMs) enable powerful applications such as sensitivity analysis, gradient‐based calibration, state estimation, boundary flux inversions, uncertainty quantification, and online machine learning. Reverse‐mode automatic differentiation (AD) efficiently provides gradients for such tasks, yet models have rarely included this capability because of complex, bespoke numerical algorithms. As part of the Differentiable programming in Julia for Earth system modeling (DJ4Earth) initiative, we present improved capabilities of the AD tool Enzyme.jl and the new compiler transpilation tool Reactant.jl, augmented by sophisticated checkpointing algorithms, which, together make general‐purpose AD tractable and efficient for full‐fledged ESM components written in Julia. Operating at the low‐level virtual machine intermediate representation or multi‐level intermediate representation compiler levels, these frameworks support mutable memory, custom kernels, and compiler optimizations before and after differentiation. Julia‐specific challenges related to just‐in‐time compilation and garbage collection are handled efficiently. Reactant further enables automatic performance portability across central processing units, graphics processing units, and tensor processing units, facilitating use of emerging AI‐customized high‐performance computing architectures. We demonstrate these frameworks on four Julia‐based ESM components featuring diverse spatial discretizations and numerical algorithms: the rotating‐sphere shallow water model ShallowWaters.jl, the finite‐volume ocean model Oceananigans.jl, the finite‐element ice sheet model DJUICE.jl, and the spectral atmospheric model SpeedyWeather.jl. Across these ESM components, our tools compute efficient and correct gradients. These results establish a foundation for differentiable, high‐performance and performance‐portable ESMs that can integrate neural networks for unresolved processes, trained online, enabling next‐generation hybrid physics–machine learning ESMs constrained by physical dynamics and observations. Plain Language Summary Earth system models are computer programs that simulate how Earth's atmosphere, ocean, ice, and biosphere interact and evolve. These models consist of millions of lines of code and rely on uncertain inputs. To improve accuracy, scientists adjust these inputs to minimize the difference between simulations and observations, measured by a “cost function.” Another computer program can efficiently determine how changes in each input affect the outcome. This calculation, called the gradient of the cost function, would be extremely time‐consuming to code manually. Instead, we use an automatic differentiation (AD) tool called Enzyme, which computes these gradients efficiently and updates them automatically whenever the model changes. As computing systems evolve rapidly, especially those optimized for artificial intelligence (AI), another tool called Reactant enables models to run efficiently across different hardware, from central processing units to graphics processing units and AI accelerators. We demonstrate these methods on four Earth system model components written in the modern programming language Julia: a shallow water model, an ocean model, an ice sheet model, and an atmospheric model. For each, the code generated via AD produces correct gradients of the cost function. This work lays the foundation for combining these differentiated models with machine learning to improve model accuracy efficiently. Key Points Four Earth system model components are successfully differentiated using the reverse mode of the automatic differentiation tool Enzyme The Julia‐based, graphics processing unit‐enabled models use bespoke numerics, with finite‐volume, finite‐element, and spectral spatial discretization schemes The compiler transpilation tool Reactant enables optimized, portable performance across diverse ML‐customized HPC architectures

JuMP is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. JuMP takes advantage of advanced features of the Julia programming language to offer unique functionality while achieving performance on par with commercial modeling tools for standard tasks. In this work we will provide benchmarks, present the novel aspects of the implementation, and discuss how JuMP can be extended to new problem classes and composed with state-of-the-art tools for visualization and interactivity.

A critical step in topology optimization (TO) is finding sensitivities. Manual derivation and implementation of sensitivities can be quite laborious and error-prone, especially for non-trivial objectives, constraints and material models. An alternate approach is to utilize automatic differentiation (AD). While AD has been around for decades, and has also been applied in TO, its wider adoption has largely been absent. In this educational paper, we aim to reintroduce AD for TO, making it easily accessible through illustrative codes. In particular, we employ JAX, a high-performance Python library for au tomatically computing sensitivities from a user-defined TO problem. The resulting framework, referred to here as AuTO, is illustrated through several examples in compliance minimization, compliant mechanism design and microstructural design.

The focus of this article is on topology optimization of heat sinks with turbulent forced convection. The goal is to demonstrate the extendibility, and the scalability of a previously developed fluid solver to coupled multi-physics and large 3D problems. The gradients of the objective and the constraints are obtained with the help of automatic differentiation applied on the discrete system without any simplifying assumptions. Thus, as demonstrated in earlier works of the authors, the sensitivities are exact to machine precision. The framework is applied to the optimization of 2D and 3D problems. Comparison between the simplified 2D setup and the full 3D optimized results is provided. A comparative study is also provided between designs optimized for laminar and turbulent flows. The comparisons highlight the importance and the benefits of full 3D optimization and including turbulence modeling in the optimization process, while also demonstrating extension of the methodology to include coupling of heat transfer with turbulent flows.

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al (2019 J. Computat. Phys. 396 483–506) for simultaneously (1) denoising the data, (2) learning and parametrizing the noise probability distribution, and (3) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

The adjoint method provides an efficient way to compute sensitivities for system models with a large number of inputs. However, implementing the adjoint method requires significant effort that limits its use. The effort is exacerbated in large-scale multidisciplinary design optimization. We propose the adoption of a three-stage compiler as the method for constructing computational models for large-scale multidisciplinary design optimization to enable accurate and efficient adjoint sensitivity analysis. We develop a new modeling language called the Computational System Design Language that provides an appropriate input to the compiler front end that works well with multidisciplinary models. This paper describes the three-stage compiler methodology and the Computational System Design Language. The proposed solution uses a graph representation of the numerical model to automatically generate a computational model that computes adjoint-based sensitivities for use within an optimization framework. For two engineering models, this approach reduces the amount of user code by a factor of approximately two compared to their original implementations, without a measurable increase in computation time. This paper also includes a best-case complexity analysis that is built into the compiler implementation to allow users to estimate the memory required to evaluate a computational model and its derivatives, which is independent of the compiler back end that ultimately generates the computational model. Future compiler implementations are expected to approach the theoretical best-case memory cost and improve run time performance for both model evaluation and derivative computation.

X‐ray fluorescence (XRF) is a popular spectroscopy technique for elemental analysis. Spectrum fitting and parameter tuning are at the core of XRF analysis and are conventionally manually intensive, especially for synchrotron experiments involving large amounts of diverse samples. This work introduces the automatic differentiation (AD) technique to XRF and an open‐source package called MapsTorch. By transforming an analytical model of the XRF spectrum into a differentiable computation graph with AD, MapsTorch enables robust optimization of parameters and elemental intensities. We evaluate MapsTorch by conducting computational experiments on a large number of historical synchrotron XRF datasets and compare its performance with the currently practiced fitting tool NLopt. The results show that MapsTorch consistently achieves high‐quality fits and often leads to better fitting quality than NLopt, particularly in tasks such as initial spectrum fitting and elemental intensity refinement. The robust performance of MapsTorch paves the way for developing automated and high‐throughput XRF data analysis workflows to handle the increasing data volumes expected from next‐generation synchrotron facilities. MapsTorch is a new open‐source package that enables automatic differentiation for X‐ray fluorescence (XRF). The robust performance of MapsTorch makes it possible to automate steps that are usually manual in XRF analysis, including initial spectrum fitting and elemental intensity refinement, and therefore can support automated, high‐throughput workflows for synchrotron facilities.