{ "types" : { "Bookmark" : { "pluralLabel" : "Bookmarks" }, "Publication" : { "pluralLabel" : "Publications" }, "GoldStandardPublication" : { "pluralLabel" : "GoldStandardPublications" }, "GoldStandardBookmark" : { "pluralLabel" : "GoldStandardBookmarks" }, "Tag" : { "pluralLabel" : "Tags" }, "User" : { "pluralLabel" : "Users" }, "Group" : { "pluralLabel" : "Groups" }, "Sphere" : { "pluralLabel" : "Spheres" } }, "properties" : { "count" : { "valueType" : "number" }, "date" : { "valueType" : "date" }, "changeDate" : { "valueType" : "date" }, "url" : { "valueType" : "url" }, "id" : { "valueType" : "url" }, "tags" : { "valueType" : "item" }, "user" : { "valueType" : "item" } }, "items" : [ { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/24864ccb34c23f7f35e3e6456d91df837/lmandl", "tags" : [ "rg-ml","myown","simliva","PN2-2","isd","EXC2075","updated" ], "intraHash" : "4864ccb34c23f7f35e3e6456d91df837", "interHash" : "02b47cd2b93c5cabeb39c9d2239071f5", "label" : "Physics-informed time-integrated DeepONet: Temporal tangent space operator learning for high-accuracy inference", "user" : "lmandl", "description" : "", "date" : "2026-03-18 08:57:48", "changeDate" : "2026-03-18 08:57:48", "count" : 3, "pub-type": "article", "journal": "Computer Methods in Applied Mechanics and Engineering", "year": "2026", "url": "https://www.sciencedirect.com/science/article/pii/S0045782526001908", "author": [ "Luis Mandl","Dibyajyoti Nayak","Tim Ricken","Somdatta Goswami" ], "authors": [ {"first" : "Luis", "last" : "Mandl"}, {"first" : "Dibyajyoti", "last" : "Nayak"}, {"first" : "Tim", "last" : "Ricken"}, {"first" : "Somdatta", "last" : "Goswami"} ], "volume": "455","pages": "118917","abstract": "Accurately modeling and inferring solutions to time-dependent partial differential equations (PDEs) over extended temporal horizons remains a core challenge in scientific machine learning. Traditional full rollout (FR) methods, predicting entire trajectories in a single pass, often fail to capture the causal dependencies inherent to dynamical systems and exhibit poor generalization outside the training time horizon. In contrast, autoregressive (AR) approaches, which evolve the system step by step, are prone to error accumulation, as predictions at each time step depend on potentially erroneous prior outputs. These shortcomings limit the long-term accuracy and reliability of both strategies. To overcome these issues, we introduce Physics-Informed Time-Integrated Deep Operator Network (PITI-DeepONet), an operator learning framework designed for stable and accurate long-term time evolution, well beyond the training time horizon. PITI-DeepONet employs a dual-output DeepONet architecture trained via either fully physics-informed or hybrid physics- and data-driven objectives. The training enforces consistency between the learned temporal derivative and its counterpart obtained via automatic differentiation. Rather than directly forecasting future states, the network learns the time-derivative operator from the current state, which is then integrated using classical time-stepping schemes - such as explicit Euler, fourth-order Runge-Kutta, second-order Adams-Bashforth-Moulton, or implicit Euler - to advance the solution sequentially in time. Additionally, the framework supports residual monitoring during inference to estimate prediction quality and flags when the learned temporal tangent becomes unreliable, e.g., outside the training domain. Applied to benchmark problems, PITI-DeepONet demonstrates enhanced accuracy and stability over extended inference time horizons when compared to traditional methods. Mean relative L2 errors reduced by 84% (versus FR) and 79% (versus AR) for the one-dimensional heat equation; by 87% (versus FR) and 98% (versus AR) for the one-dimensional Burgers equation; by 42% (versus FR) and 89% (versus AR) for the two-dimensional Allen-Cahn equation; and by 58% (vs. FR) and 61% (vs. AR) for the one-dimensional Kuramoto-Sivashinsky equation. By moving beyond classic FR and AR schemes, PITI-DeepONet paves the way for more reliable, long-term integration of complex, time-dependent PDEs.", "issn" : "0045-7825", "doi" : "https://doi.org/10.1016/j.cma.2026.118917", "bibtexKey": "MANDL2026118917" } , { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/272373d9bb43e24b26199f605ed22c33f/lmandl", "tags" : [ "qualiperf","rg-ml","myown","hybrid-mor","PN2","PN2-2","atlas","isd","updated","exc2075" ], "intraHash" : "72373d9bb43e24b26199f605ed22c33f", "interHash" : "e82eaa766ea71e21dcd0d93df23206a0", "label" : "Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning", "user" : "lmandl", "description" : "", "date" : "2024-12-03 13:57:51", "changeDate" : "2024-12-03 13:57:51", "count" : 5, "pub-type": "article", "journal": "Computer Methods in Applied Mechanics and Engineering", "year": "2025", "url": "https://www.sciencedirect.com/science/article/pii/S0045782524008405", "author": [ "Luis Mandl","Somdatta Goswami","Lena Lambers","Tim Ricken" ], "authors": [ {"first" : "Luis", "last" : "Mandl"}, {"first" : "Somdatta", "last" : "Goswami"}, {"first" : "Lena", "last" : "Lambers"}, {"first" : "Tim", "last" : "Ricken"} ], "volume": "434","pages": "117586","abstract": "The deep operator network (DeepONet) has shown remarkable potential in solving partial differential equations (PDEs) by mapping between infinite-dimensional function spaces using labeled datasets. However, in scenarios lacking labeled data, the physics-informed DeepONet (PI-DeepONet) approach, which utilizes the residual loss of the governing PDE to optimize the network parameters, faces significant computational challenges, particularly due to the curse of dimensionality. This limitation has hindered its application to high-dimensional problems, making even standard 3D spatial with 1D temporal problems computationally prohibitive. Additionally, the computational requirement increases exponentially with the discretization density of the domain. To address these challenges and enhance scalability for high-dimensional PDEs, we introduce the Separable physics-informed DeepONet (Sep-PI-DeepONet). This framework employs a factorization technique, utilizing sub-networks for individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix required for gradient computations. By incorporating forward-mode automatic differentiation (AD), we further optimize computational efficiency, achieving linear scaling of computational cost with discretization density and dimensionality, making our approach highly suitable for high-dimensional PDEs. We demonstrate the effectiveness of Sep-PI-DeepONet through three benchmark PDE models: the viscous Burgers\u2019 equation, Biot\u2019s consolidation theory, and a parameterized heat equation. Our framework maintains accuracy comparable to the conventional PI-DeepONet while reducing training time by two orders of magnitude. Notably, for the heat equation solved as a 4D problem, the conventional PI-DeepONet was computationally infeasible (estimated 289.35 h), while the Sep-PI-DeepONet completed training in just 2.5 h. These results underscore the potential of Sep-PI-DeepONet in efficiently solving complex, high-dimensional PDEs, marking a significant advancement in physics-informed machine learning.", "issn" : "0045-7825", "preprinturl" : "https://arxiv.org/abs/2407.15887", "doi" : "https://doi.org/10.1016/j.cma.2024.117586", "bibtexKey": "MANDL2025117586" } , { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/23ebccdce9fddc93484ee48096a113de2/lmandl", "tags" : [ "rg-ml","myown","simliva","PN2","PN2-2","rg-expmech-enveng","isd","EXC2075" ], "intraHash" : "3ebccdce9fddc93484ee48096a113de2", "interHash" : "d29fca572c3bd0719c2f956c5c086e66", "label" : "Comparing durability and compressive strength predictions of hyperoptimized random forests and artificial neural networks on a small dataset of concrete containing nano SiO2 and RHA", "user" : "lmandl", "description" : "", "date" : "2024-10-02 17:42:07", "changeDate" : "2024-10-02 17:42:07", "count" : 5, "pub-type": "article", "journal": "European Journal of Environmental and Civil Engineering","publisher":"Taylor & Francis", "year": "2024", "url": "", "author": [ "O. Arasteh-Khoshbin","Seyed Morteza Seyedpour","Luis Mandl","Lena Lambers","Tim Ricken" ], "authors": [ {"first" : "O.", "last" : "Arasteh-Khoshbin"}, {"first" : "Seyed Morteza", "last" : "Seyedpour"}, {"first" : "Luis", "last" : "Mandl"}, {"first" : "Lena", "last" : "Lambers"}, {"first" : "Tim", "last" : "Ricken"} ], "pages": "1\u201320", "issn" : "2116-7214", "doi" : "10.1080/19648189.2024.2393881", "bibtexKey": "ArastehKhoshbin2024" } , { "type" : "Publication", "id" : "https://puma.ub.uni-stuttgart.de/bibtex/20252f0b3fb5a37ee4a3c234a3791b9f3/lmandl", "tags" : [ "qualiperf","rg-ml","neural-network","myown","hybrid-mor","simliva","machine-learning","atlas","isd","physics-informed" ], "intraHash" : "0252f0b3fb5a37ee4a3c234a3791b9f3", "interHash" : "8596788f2da4f14c756f36923e0be9f2", "label" : "Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem", "user" : "lmandl", "description" : "", "date" : "2023-10-23 11:04:28", "changeDate" : "2024-09-25 11:21:43", "count" : 11, "pub-type": "article", "journal": "Scientific Reports", "year": "2023", "url": "https://doi.org/10.1038/s41598-023-42141-x", "author": [ "Luis Mandl","André Mielke","Seyed Morteza Seyedpour","Tim Ricken" ], "authors": [ {"first" : "Luis", "last" : "Mandl"}, {"first" : "André", "last" : "Mielke"}, {"first" : "Seyed Morteza", "last" : "Seyedpour"}, {"first" : "Tim", "last" : "Ricken"} ], "volume": "13","pages": "15566","abstract": "Physics-informed neural networks (PINNs) leverage data and knowledge about a problem. They provide a nonnumerical pathway to solving partial differential equations by expressing the field solution as an artificial neural network. This approach has been applied successfully to various types of differential equations. A major area of research on PINNs is the application to coupled partial differential equations in particular, and a general breakthrough is still lacking. In coupled equations, the optimization operates in a critical conflict between boundary conditions and the underlying equations, which often requires either many iterations or complex schemes to avoid trivial solutions and to achieve convergence. We provide empirical evidence for the mitigation of bad initial conditioning in PINNs for solving one-dimensional consolidation problems of porous media through the introduction of affine transformations after the classical output layer of artificial neural network architectures, effectively accelerating the training process. These affine physics-informed neural networks (AfPINNs) then produce nontrivial and accurate field solutions even in parameter spaces with diverging orders of magnitude. On average, AfPINNs show the ability to improve the \\$\\$\\\\backslashmathscr Ł\\\\\\_2\\$\\$relative error by \\$\\$64.84\\backslash\\%\\$\\$after 25,000 epochs for a one-dimensional consolidation problem based on Biot's theory, and an average improvement by \\$\\$58.80\\backslash\\%\\$\\$with a transfer approach to the theory of porous media.", "issn" : "2045-2322", "doi" : "10.1038/s41598-023-42141-x", "bibtexKey": "Mandl2023" } ] }