Automatic Differentiation Framework Research Articles

Performant automatic differentiation of local coupled cluster theories: Response properties and ab initio molecular dynamics

In this work, we introduce a differentiable implementation of the local natural orbital coupled cluster (LNO-CC) method within the automatic differentiation framework of the PySCFAD package. The implementation is comprehensively tuned for enhanced performance, which enables the calculation of first-order static response properties on medium-sized molecular systems using coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)]. We evaluate the accuracy of our method by benchmarking it against the canonical CCSD(T) reference for nuclear gradients, dipole moments, and geometry optimizations. In addition, we demonstrate the possibility of property calculations for chemically interesting systems through the computation of bond orders and Mössbauer spectroscopy parameters for a [NiFe]-hydrogenase active site model, along with the simulation of infrared spectra via ab initio LNO-CC molecular dynamics for a protonated water hexamer.

Dose-efficient automatic differentiation for ptychographic reconstruction

Ptychography, as a powerful lensless imaging method, has become a popular member of the coherent diffractive imaging family over decades of development. The ability to utilize low-dose X-rays and/or fast scans offers a big advantage in a ptychographic measurement (for example, when measuring radiation-sensitive samples), but results in low-photon statistics, making the subsequent phase retrieval challenging. Here, we demonstrate a dose-efficient automatic differentiation framework for ptychographic reconstruction (DAP) at low-photon statistics and low overlap ratio. As no reciprocal space constraint is required in this DAP framework, the framework, based on various forward models, shows superior performance under these conditions. It effectively suppresses potential artifacts in the reconstructed images, especially for the inherent periodic artifact in a raster scan. We validate the effectiveness and robustness of this method using both simulated and measured datasets.

End-to-End Differentiable Physics Temperature Estimation for Permanent Magnet Synchronous Motor

Differentiable physics is an approach that effectively combines physical models with deep learning, providing valuable information about physical systems during the training process of neural networks. This integration enhances the generalization ability and ensures better consistency with physical principles. In this work, we propose a framework for estimating the temperature of a permanent magnet synchronous motor by combining neural networks with the differentiable physical thermal model, as well as utilizing the simulation results. In detail, we first implement a differentiable thermal model based on a lumped parameter thermal network within an automatic differentiation framework. Subsequently, we add a neural network to predict thermal resistances, capacitances, and losses in real time and utilize the thermal parameters’ optimized empirical values as the initial output values of the network to improve the accuracy and robustness of the final temperature estimation. We validate the conceivable advantages of the proposed method through extensive experiments based on both synthetic data and real-world data and then provide some further potential applications.

Compositional Simulation for Carbon Storage in Porous Media Using an Electrolyte Association Equation of State

Summary We present a new algorithm based on automatic differentiation that enables precise computation of the derivatives of the Z-factor, facilitating the utilization of Newton’s method or coupling with a robust flow solver. Leveraging a free open-source code [MATLAB Reservoir Simulation Toolbox (MRST)], we develop an electrolyte cubic plus association (e-CPA) equation of state (EoS) model to accurately represent the injection of carbon dioxide (CO2) in brine. By integrating flow and thermodynamics, we construct an advanced compositional simulator using MRST’s object-oriented, automatic differentiation framework and the newly developed e-CPA EoS model. This simulator offers flexibility through both overall-composition and natural-variable formulations, achieved by selecting different primary variables. The Péneloux volume translation technique is employed to modify the EoS model’s volume, ensuring accurate density calculation for the mixture. Additionally, we introduce a viscosity model, e-CPA-FV, which accurately predicts the viscosity of carbon capture and storage (CCS) fluids, surpassing the accuracy of the traditional Lohrenz-Bray-Clark (LBC) model. Our simulator demonstrates superior performance in predicting CO2-brine systems compared with the standard formulation based on the Peng-Robinson (PR) EoS and can handle brine with various salts. The self-contained source code necessary to reproduce all examples is available on the open-access Zenodo digital repository (doi: 10.5281/zenodo.10691505).

4D topology optimization: Integrated optimization of the structure and self-actuation of soft bodies for dynamic motions

Topology optimization is a powerful tool utilized in various fields for structural design. However, its application has primarily been restricted to static or passively moving objects, mainly focusing on hard materials with limited deformations and contact capabilities. Designing soft and actively moving objects, such as soft robots equipped with actuators, poses challenges due to simulating dynamics problems involving large deformations and intricate contact interactions. Moreover, the optimal structure depends on the object’s motion, necessitating a simultaneous design approach. To address these challenges, we propose “4D topology optimization,” an extension of density-based topology optimization that incorporates the time dimension. This enables the simultaneous optimization of both the structure and self-actuation of soft bodies for specific dynamic tasks. Our method utilizes multi-indexed and hierarchized density variables distributed over the spatiotemporal design domain, representing the material layout, actuator layout, and time-varying actuation. These variables are efficiently optimized using gradient-based methods. Forward and backward simulations of soft bodies are done using the material point method, a Lagrangian–Eulerian hybrid approach, implemented on a recent automatic differentiation framework. We present several numerical examples of self-actuating soft body designs aimed at achieving locomotion, posture control, and rotation tasks. The results demonstrate the effectiveness of our method in successfully designing soft bodies with complex structures and biomimetic movements, benefiting from its high degree of design freedom.

Zero-order optimization for Gaussian process-based model predictive control

By enabling constraint-aware online model adaptation, model predictive control using Gaussian process (GP) regression has exhibited impressive performance in real-world applications and received considerable attention in the learning-based control community. Yet, solving the resulting optimal control problem in real-time generally remains a major challenge, due to (i) the increased number of augmented states in the optimization problem, as well as (ii) computationally expensive evaluations of the posterior mean and covariance and their respective derivatives. To tackle these challenges, we employ (i) a tailored Jacobian approximation in a sequential quadratic programming (SQP) approach and combine it with (ii) a parallelizable GP inference and automatic differentiation framework. Reducing the numerical complexity with respect to the state dimension nx for each SQP iteration from O(nx6) to O(nx3), and accelerating GP evaluations on a graphical processing unit, the proposed algorithm computes suboptimal, yet feasible, solutions at drastically reduced computation times and exhibits favorable local convergence properties. Numerical experiments verify the scaling properties and investigate the runtime distribution across different parts of the algorithm.

Reconstructing the neutron star equation of state from observational data via automatic differentiation

Neutron star observables like masses, radii, and tidal deformability are direct probes to the dense matter equation of state (EoS). A novel deep learning method that optimizes an EoS in the automatic differentiation framework of solving inverse problems is presented. The trained neural network EoS yields narrow bands for the relationship between the pressure and speed of sound as a function of the mass density. The results are consistent with those obtained from conventional approaches and the observational bound on the tidal deformability inferred from the gravitational wave event, GW170817.

Numerical simulation of fractured horizontal well considering threshold pressure gradient, non‐Darcy flow, and stress sensitivity

AbstractConventional numerical simulators cannot accurately predict the production of fractured horizontal wells in tight sandstone gas reservoirs due to the threshold pressure gradient (TPG) and stress sensitivity. Additionally, non‐Darcy flow in hydraulic fractures further complicates the prediction. Therefore, a numerical simulation model was developed to accurately predict the production of fractured horizontal wells that considers the TPG, non‐Darcy flow, and stress sensitivity using the embedded discrete fracture model. The model was solved using the automatic differentiation framework of MATLAB Reservoir Simulation Toolbox. The numerical simulation demonstrated that the Darcy flow model provided an optimistic prediction of the productivity of fractured gas wells in tight reservoirs. The TPG, non‐Darcy flow, and stress‐sensitive phenomena reduced the productivity of gas wells to varying degrees and severely affected the production of gas wells. The TPG effect, non‐Darcy effect, and stress sensitivity phenomenon reduced the gas well production by 1.66%, 12.9%, and 42.42%, respectively, compared with that of the Darcy flow. In contrast to the conventional Darcy flow model, the model established in this study can accurately predict the production of gas wells in production history matching.

Differentiable quantum chemistry with PySCF for molecules and materials at the mean-field level and beyond.

We introduce an extension to the PySCF package, which makes it automatically differentiable. The implementation strategy is discussed, and example applications are presented to demonstrate the automatic differentiation framework for quantum chemistry methodology development. These include orbital optimization, properties, excited-state energies, and derivative couplings, at the mean-field level and beyond, in both molecules and solids. We also discuss some current limitations and directions for future work.

Thermal neural networks: Lumped-parameter thermal modeling with state-space machine learning

With electric power systems becoming more compact with higher power density, the relevance of thermal stress and precise real-time-capable model-based thermal monitoring increases. Previous work on thermal modeling by lumped-parameter thermal networks (LPTNs) suffers from mandatory expert knowledge for their design and from uncertainty regarding the required power loss model. In contrast, deep learning-based temperature models cannot be designed with the low amount of model parameters as in a LPTN at equal estimation accuracy. In this work, the thermal neural network (TNN) is introduced, which unifies both, consolidated knowledge in the form of heat-transfer-based LPTNs, and data-driven nonlinear function approximation with supervised machine learning. The TNN approach overcomes the drawbacks of previous paradigms by having physically interpretable states through its state-space representation, is end-to-end differentiable through an automatic differentiation framework, and requires no material, geometry, nor expert knowledge for its design. Experiments on an electric motor data set show that a TNN achieves higher temperature estimation accuracies than previous white-/gray- or black-box models with a mean squared error of 3.18 K2 and a worst-case error of 5.84 K at 64 model parameters.

High-fidelity intensity diffraction tomography with a non-paraxial multiple-scattering model.

We propose a novel intensity diffraction tomography (IDT) reconstruction algorithm based on the split-step non-paraxial (SSNP) model for recovering the 3D refractive index (RI) distribution of multiple-scattering biological samples. High-quality IDT reconstruction requires high-angle illumination to encode both low- and high- spatial frequency information of the 3D biological sample. We show that our SSNP model can more accurately compute multiple scattering from high-angle illumination compared to paraxial approximation-based multiple-scattering models. We apply this SSNP model to both sequential and multiplexed IDT techniques. We develop a unified reconstruction algorithm for both IDT modalities that is highly computationally efficient and is implemented by a modular automatic differentiation framework. We demonstrate the capability of our reconstruction algorithm on both weakly scattering buccal epithelial cells and strongly scattering live C. elegans worms and live C. elegans embryos.

Multi-Proxy Wasserstein Classifier for Image Classification

Most widely-used convolutional neural networks (CNNs) end up with a global average pooling layer and a fully-connected layer. In this pipeline, a certain class is represented by one template vector preserved in the feature banks of fully-connected layer. Yet, a class may have multiple properties useful for recognition while the above formulation only captures one of them. Therefore, it is desired to represent a class by multiple proxies. However, directly adding multiple linear layers turns out to be a trivial solution as no improvement can be observed. To tackle this problem, we adopt optimal transport theory to calculate a non-uniform matching flow between the elements in the feature map of a sample and the proxies of a class in a closed way. By doing so, the models are enabled to achieve partial matching as both the feature maps and the proxy set can now focus on a subset of elements from the counterpart. Such formulation also enables us to embed the samples into the Wasserstein metric space, which has many advantages over the original Euclidean space. This formulation can be achieved by a lightweight iterative algorithm, which can be easily embedded into the automatic differentiation framework. Empirical studies are performed on two widely-used classification datasets, CIFAR, and ILSVRC2012, and the substantial improvements on these two benchmarks demonstrate the effectiveness of our method.

Fast optimization of parametrized quantum optical circuits

Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.

Fully implicit simulation of polymer flooding with MRST

The present work describes a fully implicit simulator for polymer injection implemented in the free, open-source MATLAB Reservoir Simulation Toolbox (MRST). Polymer injection is one of the widely used enhanced oil recovery (EOR) techniques, and complicated physical process is involved, which makes accurate simulation very challenging. The proposed work is intended for providing a powerful and flexible tool to investigate the polymer injection process in realistic reservoir scenarios. Within the model, the polymer component is assumed to be only transported in the water phase and adsorbed in the rock. The hydrocarbon phases are not influenced by the polymer, and they are described with the standard, three-phase, black-oil equations. The effects of the polymer are simulated based on the Todd–Longstaff mixing model, accounting for adsorption, inaccessible pore space, and permeability reduction effects. Shear-thinning/thickening effects based on shear rate are also included by the means of a separate inner-Newton iteration process within the global nonlinear iteration. The implementation is based on the automatic differentiation framework in MRST (MRST-AD), and an iterative linear solver with a constrained pressure residual (CPR) preconditioner is used to solve the resulting linear systems efficiently. We discuss certain implementation details to show how convenient it is to use the existing functionality in MRST to develop an accurate and efficient polymer flooding simulator for real fields. With its modular design, vectorized implementation, support for stratigraphic and general unstructured grids, and automatic differentiation framework, MRST is a very powerful prototyping and experimentation platform for development of new reservoir simulators. To verify the simulator, we first compare it with a commercial simulator and good agreement is achieved. Then, we apply the new simulator to a few realistic reservoir models to investigate the effect of adding polymer injection, and computational efficiency is demonstrated. Finally, we combine existing optimization functionality in MRST with the new polymer simulator to optimize polymer flooding for two different reservoir models. We argue that the presented software framework can be used as an efficient prototyping tool to evaluate new models for polymer–water flooding processes in real reservoir fields.

A backward automatic differentiation framework for reservoir simulation

In numerical reservoir simulations, Newton’s method is a concise, robust and, perhaps the most commonly used method to solve nonlinear partial differential equations (PDEs). However, as reservoir simulators incorporate more and more physical and chemical phenomena, writing codes that compute gradients for reservoir simulation equations can become quite complicated. This paper presents an automatic differentiation (AD) framework that is specially designed for simplifying coding and simultaneously maintaining computational efficiency. First a parse tree for a mathematical expression is built and evaluated with the backward mode AD, and then the derivatives with respect to the expression’s arguments are transformed to derivatives with respect to the PDE’s independent variables. The first stage can be realized either by runtime polymorphism to gain higher flexibility or by compile-time polymorphism to gain faster execution speed; the second stage is realized by linear combinations of sparse vectors, which can be accelerated by recording the target column indices. The AD framework has been implemented in an in-house reservoir simulator. Individual tests on some complex mathematical expressions were carried out to compare the speed of the manual implementation, the runtime polymorphic implementation and the compile-time polymorphic implementation of the differentiation. Then the performance of the three was analyzed in complete simulations. These cases indicate that the proposed approach has good efficiency and is applicable to reservoir simulations.