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- Solving steady-state elliptic problems in irregular domains using physics-informed neural networks and fictitious domain methodsPublication . Rodrigues, José; Rodrigues, José AlbertoThis paper introduces an innovative methodology for solving steady-state elliptic partial differential equations defined over irregular domains, by coupling the capabilities of Physics-Informed Neural Networks with the Fictitious Domain Method. The primary emphasis is placed on applications involving the heat equation, a fundamental model in thermal analysis where the complexity of non-standard geometries often poses significant challenges for traditional numerical methods. The proposed approach exploits the inherent strength of Physics-Informed Neural Networks in embedding the underlying physical laws directly into the learning process, enabling the model to approximate solutions without relying on meshbased discretization. Simultaneously, the Fictitious Domain Method facilitates the treatment of irregular computational domains by embedding them within a larger, regular domain, thereby simplifying the application of boundary conditions and numerical operations. The synergy between these two techniques results in a flexible, efficient, and accurate computational framework that is well-suited for addressing heat transfer problems in complex geometrical configurations.
- Experimental study on drag coefficient of flexible vegetation under non-breaking wavesPublication . Reis, A. Rui; Fortes, Conceição J. E. M.; ; Rodrigues, José Alberto; Hu, Zhan; Suzuki, TomohiroLaboratory experiments of wave propagation over rigid and flexible vegetation fields, with the same configurations, were conducted to understand the effect of vegetation flexibility on the drag coefficient (CD). The direct method and the least squares method (LSM), based on force and flow measurements, are applied to calculate the CD in the xperimental conditions. The formulations of both methods are extended to estimate the CD for flexible vegetation cases. A video analysis was performed to account for the swaying motion. Typically, wave dissipation is lower for flexible than for rigid vegetation of the same configuration, under the same flow condition. Therefore, a proportional effect in the corresponding CD results, obtained from common CD calibration to wave dissipation without considering vegetation motion, is usually observed. However, the present results show that although the wave dissipation was 34% lower for flexible relative to rigid vegetation, the respective CD values were close. CD estimations considering vegetation motion and inertia suggest that CD of flexible vegetation was up to 13% higher relative to rigid vegetation. Accounting for inertia reduced the CD for rigid vegetation up to 7%, while raised the CD for flexible vegetation up to 13%.
- Using physics-informed neural networks (PINNs) for tumor cell growth modelingPublication . Rodrigues, JoséThis paper presents a comprehensive investigation into the applicability and performance of two prominent growth models, namely, the Verhulst model and the Montroll model, in the context of modeling tumor cell growth dynamics. Leveraging the power of Physics-Informed Neural Networks (PINNs), we aim to assess and compare the predictive capabilities of these models against experimental data obtained from the growth patterns of tumor cells. We employed a dataset comprising detailed measurements of tumor cell growth to train and evaluate the Verhulst and Montroll models. By integrating PINNs, we not only account for experimental noise but also embed physical insights into the learning process, enabling the models to capture the underlying mechanisms governing tumor cell growth. Our findings reveal the strengths and limitations of each growth model in accurately representing tumor cell proliferation dynamics. Furthermore, the study sheds light on the impact of incorporating physics-informed constraints on the model predictions. The insights gained from this comparative analysis contribute to advancing our understanding of growth models and their applications in predicting complex biological phenomena, particularly in the realm of tumor cell proliferation.
- A finite element method for shear stress study on cancer cell proliferationPublication . Rodrigues, José; Rodrigues, José AlbertoMetastatic progression of tumors requires the coordinated dissemination of cancerous cells through interstitial tissues and their replication in distant body locations. Despite their importance in cancer treatment decisions, key factors, such as cell shape adaptation and the role it plays in dense tissue invasion by cancerous cells, are not well understood. With this work, we will continue further work on the study of the effect of shear stress on cancer cell proliferation. We present a coupled Keller-Segel and Darcy–Brinkman model implemented in finite elements to obtain numerical results.
- Trapped modes along periodic structures submerged in a three-layer fluid with a background steady flowPublication . Dias, Gonçalo; Pereira, Bruno M. M.In this study, we study the trapping of linear water waves by infinite arrays of three-dimensional fixed periodic structures in a three-layer fluid. Each layer has an independent uniform velocity field with respect to the fixed ground in addition to the internal modes along the interfaces between layers. Dynamical stability between velocity shear and gravitational pull constrains the layer velocities to a neighbourhood of the diagonal U1=U2=U3 in velocity space. A non-linear spectral problem results from the variational formulation. This problem can be linearized, resulting in a geometric condition (from energy minimization) that ensures the existence of trapped modes within the limits set by stability. These modes are solutions living the discrete spectrum that do not radiate energy to infinity. Symmetries reduce the global problem to solutions in the first octant of the three-dimensional velocity space. Examples are shown of configurations of obstacles which satisfy the stability and geometric conditions, depending on the values of the layer velocities. The robustness of the result of the vertical column from previous studies is confirmed in the new configurations. This allows for comparison principles (Cavalieri's principle, etc.) to be used in determining whether trapped modes are generated.
- Stochastic graph-based models of tumor growth and celular interationsPublication . Rodrigues, José AlbertoThe tumor microenvironment is a highly dynamic and complex system where cellular interactions evolve over time, influencing tumor growth, immune response, and treatment resistance. In this study, we develop a graph-theoretic framework to model the tumor microenvironment, where nodes represent different cell types, and edges denote their interactions. The temporal evolution of the tumor microenvironment is governed by fundamental biological processes, including proliferation, apoptosis, migration, and angiogenesis, which we model using differential equations with stochastic effects. Specifically, we describe tumor cell population dynamics using a logistic growth model incorporating both apoptosis and random fluctuations. Additionally, we construct a dynamic network to represent cellular interactions, allowing for an analysis of structural changes over time. Through numerical simulations, we investigate how key parameters such as proliferation rates, apoptosis thresholds, and stochastic fluctuations influence tumor progression and network topology. Our findings demonstrate that graph theory provides a powerful mathematical tool to analyze the spatiotemporal evolution of tumors, offering insights into potential therapeutic strategies. This approach has implications for optimizing cancer treatments by targeting critical network structures within the tumor microenvironment.
- Flow of a periodic interfacial travelling water wavePublication . Cal, Filipe; Dias, GonçaloWe consider a symmetric periodic travelling wave propagating at the interface between two homogeneous, incompressible, irrotational and inviscid fluids bounded by horizontal planes. For interfacial waves of small amplitude, we present a formula for the interface wave depending on the pressure at the rigid lid and at the flat bottom, and, for the general non-linear case, we derive a lower bound for the interfacial wave height. Under certain conditions imposed on the horizontal component of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we study the monotonicity of the horizontal component of the velocity field along the streamlines and also analyze the monotonicity of the pressure along horizontal lines throughout the fluid in both layers, and along the boundary of the domain, between the crest and the trough. Finally, based on the behavior of the velocity field components, we build a pictorial description of the particle paths in both layers.
- Trapped modes along periodic structures submerged in a two-layer fluid with free surface and a background steady flowPublication . Dias, Gonçalo; Pereira, BrunoThis study examines the trapping of linear water waves by an endless structure of stationary, three-dimensional periodic obstacles within a two-layer fluid system. The setup features a lower layer of either limited or unlimited depth, overlaid by an upper layer of finite thickness bounded by a free surface, with each layer exhibiting its own constant background speed relative to the fixed reference frame. For real roots to emerge in the dispersion relation, an additional stability condition on the layer velocities is necessary. By selecting adequate choices for the background flow, a non-linear eigenvalue problem is derived from the variational formulation; its reasonable approximation yields a geometric criterion that guarantees the presence of trapped modes (subject to the aforementioned stability bounds). The selection of the eigenvalue is influenced by velocity owing to the presence of an interface and free surface. Due to inherent symmetries, the overall analysis can be confined to the positive quadrant of the velocity domain. Illustrations are provided for various obstacle setups that produce trapped modes in diverse ways.
- Error estimates and generalized trial constructions for solving ODEs using physics-informed neural networksPublication . Babni, Atmane; Jamiai, Ismail; Rodrigues, José AlbertoIn this paper, we address the challenge of solving differential equations using physics-informed neural networks (PINNs), an innovative approach that integrates known physical laws into neural network training. The PINN approach involves three main steps: constructing a neural-network-based solution ansatz, defining a suitable loss function, and minimizing this loss via gradient-based optimization. We review two primary PINN formulations: the standard PINN I and an enhanced PINN II. The latter explicitly incorporates initial, final, or boundary conditions. Focusing on first-order differential equations, PINN II methods typically express the approximate solution as u˜(x,θ)=P(x)+Q(x)N(x,θ), where N(x,θ) is the neural network output with parameters θ, and P(x) and Q(x) are polynomial functions. We generalize this formulation by replacing the polynomial Q(x) with a more flexible function ϕ(x). We demonstrate that this generalized form yields a uniform approximation of the true solution, based on Cybenko’s universal approximation theorem. We further show that the approximation error diminishes as the loss function converges. Numerical experiments validate our theoretical findings and illustrate the advantages of the proposed choice of ϕ(x). Finally, we outline how this framework can be extended to higher-order or other classes of differential equations.
- Fractal Laplace transform: analyzing fractal curvesPublication . Khalili Golmankhaneh, Alireza; Welch, Kerri; Serpa, Cristina; Rodríguez-Lopez, RosanaThe concept of Laplace transform has been extended to fractal curves, enabling the solution of fractal differential equations with constant coefficients. This extension, known as the fractal Laplace transform, is particularly useful for handling inhomogeneous differential equations that involve delta Dirac functions and step functions within the realm of fractal functions. A comprehensive table of essential formulas for the fractal Laplace transform has been compiled to facilitate its application in various scenarios. By utilizing this transformative approach, researchers can now delve into the study of fractal functions and address complex problems involving non-traditional geometries. To illustrate the practicality of the fractal Laplace transform, several examples are provided, showcasing its effectiveness in solving fractal differential equations. This advancement represents a significant augmentation of the classical Laplace transform, tailored to suit the distinctive characteristics of fractal systems and functions.