Deep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the perydinamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.

We study the implementation of a Chebyshev spectral method with forward Euler integrator proposed in Berardi et al.(2023) to investigate a peridynamic nonlocal formulation of Richards’ equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.

In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carathéodory type right-hand side functions. We provide construction of the randomized Euler scheme for DDEs and investigate its error. We also report results of numerical experiments.

Physics-Informed Neural Networks (PINN) are a machine learning tool that can be used to solve direct and inverse problems related to models described by Partial Differential Equations by including in the cost function to minimise during training the residual of the differential operator. This paper proposes an adaptive inverse PINN applied to different transport models, from diffusion to advection–diffusion–reaction, and mobile–immobile transport models for porous materials. Once a suitable PINN is established to solve the forward problem, the transport parameters are added as trainable parameters and the reference data is added to the cost function. We find that, for the inverse problem to converge to the correct solution, the different components of the loss function (data misfit, initial conditions, boundary conditions and residual of the transport equation) need to be weighted adaptively as a function of the training iteration (epoch). Similarly, gradients of trainable parameters are scaled at each epoch accordingly. Several examples are presented for different test cases to support our PINN architecture and its scalability and robustness.

We discuss the dynamics of a (neutral) test particle in topological star spacetime undergoing scattering processes by a superposed test radiation field, a situation that in a 4D black hole spacetime is known as relativistic Poynting-Robertson effect, paving the way for future studies involving radiation-reaction effects. Furthermore, we study self-force-driven evolution of a scalar field, perturbing the top-star spacetime with a scalar charge current. The latter for simplicity is taken to be circular, equatorial and geodetic. To perform this study, besides solving all the self-force related problem (regularization of all divergences due to the self-field, mode sum regularization, etc.), we had to adapt the 4D Mano-Suzuki-Takasugi formalism to the present 5D situation. Finally, we have compared this formalism with the (quantum) Seiberg-Witten formalism, both of which are related to the solutions of a Heun confluent equation but appear in different contexts in the literature: the first in black hole perturbation theory and the second in quantum curves in super-Yang-Mills theories.

Using the multipolar post-Minkowskian formalism, we compute the frequency-domain waveform generated by the gravitational scattering of two nonspinning bodies at the fourth post-Minkowskian order (O(G4), or two-loop order), and at the fractional second post-Newtonian accuracy [O(v4/c4)]. The waveform is decomposed in spin-weighted spherical harmonics and the needed radiative multipoles, Um(ω),Vm(ω), are explicitly expressed in terms of a small number of master integrals. The basis of master integrals contains both (modified) Bessel functions, and solutions of inhomogeneous Bessel equations with Bessel-function sources. We show how to express the latter in terms of Meijer G functions. The low-frequency expansion of our results is checked against existing classical soft theorems. We also complete our previous results on the O(G2) bremsstrahlung waveform by computing the O(G3) spectral densities of radiated energy and momentum, in the rest frame of one body, at the thirtieth order in velocity.

In this paper we present an approach to compute analytical post-Minkowskian corrections to unbound two-body scattering in the self-force formalism. Our method relies on a further low-velocity (post-Newtonian) expansion of the motion. We present a general strategy valid for gravitational and nongravitational self-force, and we explicitly demonstrate our approach for a scalar charge scattering off a Schwarzschild black hole. We compare our results with recent calculations in [L. Barack, Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model, Phys. Rev. D 108, 024025 (2023)PRVDAQ2470-001010.1103/PhysRevD.108.024025], showing complete agreement where appropriate and fixing undetermined scale factors in their calculation. Our results also extend their results by including in our dissipative sector the contributions from the flux into the black hole horizon.

We compute the purely local-in-time (scale-free and logarithm-free) part of the conservative dynamics of gravitationally interacting two-body systems at the fourth post-Minkowskian order, and at the thirtieth order in velocity. The gauge-invariant content of this fourth post-Minkowskian local dynamics is given in two ways: (i) its contribution to the on-shell action (for both hyperboliclike and ellipticlike motions); and (ii) its contribution to the effective one-body Hamiltonian (in energy gauge). Our computation capitalizes on the tutti frutti approach [D. Bini, Novel approach to binary dynamics: Application to the fifth post-Newtonian level, Phys. Rev. Lett. 123, 231104 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.231104] and on recent post-Minkowskian advances [Z. Bern, Scattering amplitudes, the tail effect, and conservative binary dynamics at O(G4), Phys. Rev. Lett. 128, 161103 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.161103; C. Dlapa, Conservative dynamics of binary systems at fourth post-Minkowskian order in the large-eccentricity expansion, Phys. Rev. Lett. 128, 161104 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.161104; C. Dlapa, Local in time conservative binary dynamics at fourth post-Minkowskian order, 132, 221401 (2024)PRLTAO0031-900710.1103/PhysRevLett.132.221401].

We revisit the quantum-amplitude-based derivation of the gravitational waveform emitted by the scattering of two spinless massive bodies at the third order in Newton's constant, h∼G+G2+G3 (one-loop level), and correspondingly update its comparison with its classically derived multipolar-post-Minkowskian counterpart. A spurious-pole-free reorganization of the one-loop five-point amplitude substantially simplifies the post-Newtonian expansion. We find complete agreement between the two results up to the fifth order in the small velocity expansion after taking into account three subtle aspects of the amplitude derivation: (1) in agreement with [A. Georgoudis et al., J. High Energy Phys. 03 (2024) 08910.1007/JHEP03(2024)089], the term quadratic in the amplitude in the observable-based formalism [D. A. Kosower et al., J. High Energy Phys. 02 (2019) 137JHEPFG1029-847910.1007/JHEP02(2019)137] generates a frame rotation by half the classical scattering angle; (2) the dimensional regularization of the infrared divergences of the amplitude introduces an additional (d-4)/(d-4) finite term; and (3) zero-frequency gravitons are found to contribute additional terms both at order h∼G1 and at order h∼G3 when including disconnected diagrams in the observable-based formalism.

We investigate the geometrical properties, spectral classification, geodesics, and causal structure of Bonnor's spacetime [W. B. Bonnor, A rotating dust cloud in general relativity, J. Phys. A 10, 1673 (1977)JPHAC50305-447010.1088/0305-4470/10/10/004], i.e., a stationary axisymmetric solution with a rotating dust as a source. This spacetime has a directional singularity at the origin of the coordinates (related to the diverging vorticity field of the fluid there), which is surrounded by a toroidal region where closed timelike curves (CTCs) are allowed, leading to chronology violations. We use the effective potential approach to provide a classification of the different kind of geodesic orbits on the symmetry plane as well as to study the helical-like motion around the symmetry axis on a cylinder with constant radius. In the former case we find that, as a general feature for positive values of the angular momentum, test particles released from a fixed space point and directed toward the singularity are repelled and scattered back as soon as they approach the CTC boundary, without reaching the central singularity. In contrast, for negative values of the angular momentum there exist conditions in the parameter space for which particles are allowed to enter the pathological region. Finally, as a more realistic mechanism, we study accelerated orbits undergoing friction forces due to the interaction with the background fluid, which may also act in order to prevent particles from approaching the CTC region.

We discuss the solution proposed by Fermi to the so called “4/3 problem” in the classical theory of the electron, a problem which puzzled the physics community for many decades before and after his contribution. Unfortunately his early resolution of the problem in 1922–1923 published in three versions in Italian and German journals (after three preliminary articles on the topic) went largely unnoticed. Even more recent texts devoted to classical electron theory still do not present his argument or acknowledge the actual content of those articles. The calculations initiated by Fermi at the time are completed here by formulating and discussing the conservation of the total 4-momentum of the accelerated electron as seen from the instantaneous rest frame in which it is momentarily at rest.

The flow of emulsions in confined microfluidic channels is affected by surface roughness. Directional roughness effects have recently been reported in channels with asymmetric boundary conditions featuring a flat wall, and a wall textured with directional roughness, the latter promoting a change in the velocity profiles when the flow direction of emulsions is inverted [D. Filippi et al., Adv. Mater. Technol., 2023, 8, 2201748]. An operative protocol is needed to reconstruct the stress profile inside the channel from velocity data to shed light on the trigger of the directional response. To this aim, we performed lattice Boltzmann numerical simulations of the flow of model emulsions with a minimalist model of directional roughness in two dimensions: a confined microfluidic channel with one flat wall and the other patterned by right-angle triangular-shaped posts. Simulations are essential to develop a protocol based on mechanical arguments to reconstruct stress profiles. Hence, one can analyze data to relate directional effects in velocity profiles to different rheological responses close to the rough walls associated with opposite flow directions. We finally show the universality of this protocol by applying it to other realizations of directional roughness by considering experimental data on emulsions in a microfluidic channel featuring a flat wall and a wall textured by herringbone-shaped roughness.

The steady-state behavior of a single drop under shear flow has been extensively investigated in the limit of small deformation and negligible inertia effects. In this work, we combine the calculations proposed by Flumerfelt [R. W. Flumerfelt, J. Colloid Interface Sci. 76, 330 (1980)0021-979710.1016/0021-9797(80)90377-X] for unconfined drops with interface viscosity, with those by Shapira and Haber [M. Shapira and S. Haber, Int. J. Multiphase Flow 16, 305 (1990)0301-932210.1016/0301-9322(90)90061-M] for confined drops without interface viscosity. By merging these two approaches, we provide comprehensive analytical predictions for steady-state drop deformation and inclination angle across a wide range of physical conditions, from confined to unconfined droplets, including or excluding the effect of interface viscosity. The proposed analytical predictions are also robust concerning variations in the viscosity ratio, making our model general enough to include any of the above conditions.

We study the regularity properties of the weak solutions u : Ω ⊆ Rn → R to elliptic problems −div a(x,Du) + b(x)φ′(|u|) u |u| = f in Ω , u = 0 on ∂Ω , with Ω ⊂ Rn a bounded open set and where the function a(x, ξ) satisfies growth conditions with respect to the second variable expressed through an N-function φ. We prove that, under a suitable interplay between the lower order terms and the datum f, which is assumed only to belong to L1(Ω), the solutions are bounded in Ω. Next, if a(x, ξ) depends on x through a H ̈older continuous function we take advantage from the boundedness of the solution u to prove the higher differentiability and the higher integrability of its gradient, under mild assumptions on the data.

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on Rn to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

The reciprocal energy and enstrophy transfers between normal fluid and superfluid components dictate the overall dynamics of superfluid 4He including the generation, evolution and coupling of coherent structures, the distribution of energy among lengthscales, and the decay of turbulence. To better understand the essential ingredients of this interaction, we employ a numerical two-way model which self-consistently accounts for the back-reaction of the superfluid vortex lines onto the normal fluid. Here we focus on a prototypical laminar (non-turbulent) vortex configuration which is simple enough to clearly relate the geometry of the vortex line to energy injection and dissipation to/from the normal fluid: a Kelvin wave excitation on two vortex anti-vortex pairs evolving in (a) an initially quiescent normal fluid, and (b) an imposed counterflow. In (a), the superfluid injects energy and vorticity in the normal fluid. In (b), the superfluid gains energy from the normal fluid via the Donnelly-Glaberson instability.

In this paper, we study the influence of the boundary conditions of the velocity fields in superfluid helium counterflow experiments. To make progress, we perform numerical simulations where we allow a slip velocity of the viscous component at the walls, and observe how this impacts on velocity fields and density profiles of distribution of quantized vortices. We conclude that the presence of a slip velocity at the walls generates a more homogeneous vortex distribution throughout the channel.

The REDRAW project investigates the exploitation of the federated learning computing paradigm to improve the technologies adopted for the monitoring, diagnosis and treatment management of specific health conditions, developing approaches more respectful of the constraints of privacy, confidentiality and cybersecurity, which are still largely absent from the market. REDRAW proposes the study and fine-tuning of dynamic cloud-edge deployment techniques, which exploits Federated Learning (FL) models, in three real-world contexts, to improve the technological features of existing solutions, while respecting the strategic and non-functional constraints that characterize the Italian and European scenarios .

Multiple sclerosis (MS) is a demyelinating neurological disorder with a highly heterogeneous clinical presentation and course of progression. Disease-modifying therapies are the only available treatment, as there is no known cure for the disease. Careful selection of suitable therapies is necessary, as they can be accompanied by serious risks and adverse effects such as infection. Magnetic resonance imaging (MRI) plays a central role in the diagnosis and management of MS, though MRI lesions have displayed only moderate associations with MS clinical outcomes, known as the clinico-radiological paradox. With the advent of machine learning (ML) in healthcare, the predictive power of MRI can be improved by leveraging both traditional and advanced ML algorithms capable of analyzing increasingly complex patterns within neuroimaging data. The purpose of this review was to examine the application of MRI-based ML for prediction of MS disease progression. Studies were divided into five main categories: predicting the conversion of clinically isolated syndrome to MS, cognitive outcome, EDSS-related disability, motor disability and disease activity. The performance of ML models is discussed along with highlighting the influential MRI-derived biomarkers. Overall, MRI-based ML presents a promising avenue for MS prognosis. However, integration of imaging biomarkers with other multimodal patient data shows great potential for advancing personalized healthcare approaches in MS.

Medical digital twins are computational models of human biology relevant to a given medical condition, which are tailored to an individual patient, thereby predicting the course of disease and individualized treatments, an important goal of personalized medicine. The immune system, which has a central role in many diseases, is highly heterogeneous between individuals, and thus poses a major challenge for this technology. In February 2023, an international group of experts convened for two days to discuss these challenges related to immune digital twins. The group consisted of clinicians, immunologists, biologists, and mathematical modelers, representative of the interdisciplinary nature of medical digital twin development. A video recording of the entire event is available. This paper presents a synopsis of the discussions, brief descriptions of ongoing digital twin projects at different stages of progress. It also proposes a 5-year action plan for further developing this technology. The main recommendations are to identify and pursue a small number of promising use cases, to develop stimulation-specific assays of immune function in a clinical setting, and to develop a database of existing computational immune models, as well as advanced modeling technology and infrastructure.