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DOE OSTI2023

Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

A continuous-time dynamical system with parameter ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as ε approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Code Demonstration: Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

In this notebook, we use the symplectic gyroceptron architecture from [Duruisseaux et al., 2022] to learn a surrogate map for the nearly-periodic symplectic flow map associated to a nearly-periodic Hamiltonian system composed of two nonlinearly coupled oscillators, where one of them oscillates significantly faster than the other: $\Big\lbrace$ $ {\dot{q}_1 = p_1 \hspace{5mm} \dot{p}_1 = -q_1 - \epsilon \partial_{q1}U(q_1 , q_2 )} \atop {\dot{q}_2 = \epsilon p_2 \hspace{4mm} \dot{p}_2 = -\epsilon q_2 - \epsilon \partial_{q2}U(q_1 , q_2 )}$. These equations of motion are the Hamilton's equations associated to the Hamiltonian $H_{\epsilon}(q_1,q_2,p_1,p_2)=\frac{1}{2}(q^2_1+p^2_1)+\frac{1}{2}\epsilon(q^2_2+p^2_2)+\epsilon$$U(q_1,q_2)$.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A Method for Dimensionally Adaptive Sparse Trigonometric Interpolation of Periodic Functions

We present a method for dimensionally adaptive sparse trigonometric interpolation of multidimensional periodic functions belonging to a smoothness class of finite order. This method targets applications where periodicity must be preserved and the precise anisotropy is not known a priori. To the authors' knowledge, this is the first instance of a dimensionally adaptive sparse interpolation algorithm that uses a trigonometric interpolation basis. The motivating application behind this work is the adaptive approximation of a multi-input model for a molecular potential energy surface (PES) where each input represents an angle of rotation. Our method is based on an anisotropic quasi-optimal estimate for the decay rate of the Fourier coefficients of the model; a least-squares fit to the coefficients of the interpolant is used to estimate the anisotropy. Thus, our adaptive approximation strategy begins with a coarse isotropic interpolant, which is gradually refined using the estimated anisotropic rates. The procedure takes several iterations where ever-more accurate interpolants are used to generate ever-improving anisotropy rates. We present several numerical examples of our algorithm where the adaptive procedure successfully recovers the theoretical “best” convergence rate, including an application to a periodic PES approximation. An open-source implementation of our algorithm resides in the Tasmanian UQ library developed at Oak Ridge National Laboratory.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1978

Forced oscillations in quadratically damped systems

Bayliss (1975) has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations. In the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear. The properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution. By using the Lipschitz continuity of the mapping and the contracting mapping principle, almost periodic solutions can be found for perturbations of the considered equation. Attention is also given to the Lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory.

Bayliss, A.↗

NASA NTRS1989

Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques

A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated.

Banks, H. T.↗

NASA NTRS1989

Optimal feedback control of infinite-dimensional parabolic evolution systems - Approximation techniques

Banks, H. T.↗

NASA NTRS1983

Structure and other properties of Jupiter's distant magnetotail

Analyses of data from Voyager 2 experiments provide evidence for, and characteristics of, a Jovian magnetotail extending at least to 9,000 Jovian radii from the planet. During approximately (25 day) periodic sightings of the tail, the magnetic field tended to point radially towards or away from Jupiter, indicating preservation to large distances of the bipolar, lobe like structure observed near the planet. This periodicity, along with various properties of the solar wind at this time, indicates that the tail is apparently influenced by recurrent solar wind features. Anomalous magnetic fields, not aligned with the nominal tail axis, also exist within the tail, especially in the low density, central (core) region, indicating some complexity of internal structure.

Lepping, R. P.↗

NASA NTRS1983

Structure and other properties of Jupiter's distant magnetotail

Lepping, R. P.↗

DOE OSTI2025

Pulse: An Outlier Sensitive Downsampling Algorithm For Timeseries Data

Pulse is a downsampling algorithm for timeseries data. Frequently datasets become so large that visualization tools and web browsers cannot effectively render graphics due to memory constraints. Downsampling algorithms are commonly applied to minimize the quantity of data required to visualize important features or trends in the data, but some datasets are composed by distinct enough features and trends that most existing downsampling algorithms fail to preserve them. Pule was developed to downsample timeseries data for galvanostatic stack test data at the Idaho National Laboratory. These datasets were composed by approximately 4 million records, most of them being extremely uniform. However, during relatively brief time periods when the stack test changes state, for example when the test article is powered on, or a load is added, the data produce sparse asymptotes. No existing downsampling algorithm was capable of preserving the sparse asymptotes in electrolysis stack test data. Instead, we develop a downsampling algorithm that preserves important outliers in data, and otherwise aggressively downsamples uniform data. The algorithm has applications in other domains like seismology, in the measurement of earthquakes, or astronomy, in the measurement of quasars or transit photometry.

Woodruff, Nathan [Idaho National Laboratory (INL),↗

DOE OSTI2024

Coarse-graining Hamiltonian systems using WSINDy

Abstract Weak form equation learning and surrogate modeling has proven to be computationally efficient and robust to measurement noise in a wide range of applications including ODE, PDE, and SDE discovery, as well as in coarse-graining applications, such as homogenization and mean-field descriptions of interacting particle systems. In this work we extend this coarse-graining capability to the setting of Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth $$\varepsilon$$ ε -dependent Hamiltonian vector field $$X_\varepsilon$$ X ε possesses an approximate symmetry if the limiting vector field $$X_0=\lim _{\varepsilon \rightarrow 0}X_\varepsilon$$ X 0 = lim ε → 0 X ε possesses an exact symmetry. Such approximate symmetries often lead to the existence of a Hamiltonian system of reduced dimension that may be used to efficiently capture the dynamics of the symmetry-invariant dependent variables. Deriving such reduced systems, or approximating them numerically, is an ongoing challenge. We demonstrate that WSINDy can successfully identify this reduced Hamiltonian system in the presence of large perturbations imparted in the $$\varepsilon >0$$ ε > 0 regime, while remaining robust to extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. The methodology is computationally efficient, often requiring only a single trajectory to learn the global reduced Hamiltonian, and avoiding forward solves in the learning process. In this way, we argue that weak-form equation learning is particularly well-suited for Hamiltonian coarse-graining. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order system, with dimension reduced by at least two, upon observation of the relevant degrees of freedom. While our main contribution is computational, we also provide a contribution to the literature on averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems. This provides theoretical justification for our approach as WSINDy’s computations occur at the level of Hamiltonian vector fields. We illustrate the efficacy of our proposed method using physically relevant examples, including coupled oscillator dynamics, the Hénon–Heiles system for stellar motion within a galaxy, and the dynamics of charged particles.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1986

Trace elements in Antarctic meteorites: Weathering and genetic information

Antarctic meteorite discoveries have created great scientific interest due to the large number of specimens recovered (approximately 7000) and because included are representatives of hitherto rare or unknown types. Antarctic meteorites are abundant because they have fallen over long periods and were preserved, transported, and concentrated by the ice sheets. The weathering effects on the Antarctic meteorites are described. Weathering effects of trace element contents of H5 chondrites were studied in detail. The results are examined. The properties of Antarctic finds and non-Antarctic falls are discussed.

Lipschutz, M. E.↗

DOE OSTI2023

Physics informed deep neural network embedded in a chemical transport model for the Amazon rainforest

Secondary organic aerosols (SOA) are fine particles in the atmosphere, which interact with clouds, radiation and affect the Earth’s energy budget. SOA formation involves chemistry in gas phase, aqueous aerosols, and clouds. Simulating these chemical processes involve solving a stiff set of differential equations, which are computationally expensive steps for three-dimensional chemical transport models. Deep neural networks (DNNs) are universal function approximators that could be used to represent the complex nonlinear changes in aerosol physical and chemical processes; however, key challenges such as generalizability to extended time periods, preservation of mass balance, simulating sparse model outputs, and maintaining physical constraints have limited their use in atmospheric chemistry. Here, we develop an approach of using a physics-informed DNN that overcomes previous such challenges and demonstrates its applicability for the chemical formation processes of isoprene epoxydiol SOA (IEPOX-SOA) over the Amazon rainforest. The DNN is trained with data generated by simulating IEPOX-SOA over the entire atmospheric column, using the Weather Research and Forecasting Model coupled with Chemistry (WRF-Chem). The trained DNN is then embedded within WRF-Chem to replace the computationally expensive default solver of IEPOX-SOA formation. The trained DNN predictions generalizes well with the default model simulation of the IEPOX-SOA mass concentrations and its size distribution (20 size bins) over several days of simulations in both dry and wet seasons. The embedded DNN reduces the computational expense of WRF-Chem by a factor of 2. Our approach shows promise in terms of application to other computationally expensive chemistry solvers in climate models.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2007

Iapetus' Geophysics: Rotation Rate, Shape, and Equatorial Ridge

Iapetus has preserved evidence that constrains the modeling of its geophysical history from the time of its accretion until now. The evidence is (a) its present 79.33-day rotation or spin rate, (b) its shape that corresponds to the equilibrium figure for a hydrostatic body rotating with a period of approximately 16 h, and (c) its high, equatorial ridge, which is unique in the Solar System. This paper reports the results of an investigation into the coupling between Iapetus' thermal and orbital evolution for a wide range of conditions including the spatial distributions with time of composition, porosity, short-lived radioactive isotopes (SLRI), and temperature. The thermal model uses conductive heat transfer with temperature-dependent conductivity. Only models with a thick lithosphere and an interior viscosity in the range of about the water ice melting point can explain the observed shape. Short-lived radioactive isotopes provide the heat needed to decrease porosity in Iapetus? early history. This increases thermal conductivity and allows the development of the strong lithosphere that is required to preserve the 16-h rotational shape and the high vertical relief of the topography. Long-lived radioactive isotopes and SLRI raise internal temperatures high enough that significant tidal dissipation can start, and despin Iapetus to synchronous rotation. This occurred several hundred million years after Iapetus formed. The models also constrain the time when Iapetus formed because the successful models are critically dependent upon having just the right amount of heat added by SLRI decay in this early period. The amount of heat available from short-lived radioactivity is not a free parameter but is fixed by the time when Iapetus accreted, by the canonical concentration of Al-26, and, to a lesser extent, by the concentration of Fe-60. The needed amount of heat is available only if Iapetus accreted between 2.5 and 5.0Myr after the formation of the calcium aluminum inclusions as found in meteorites. Models with these features allow us to explain Iapetus? present synchronous rotation, its fossil 16-h shape, and the context within which the equatorial ridge arose.

geophysics↗

DOE OSTI2025

A Practical Framework for Simulating Time-Resolved Spectroscopy Based on a Real-Time Dyson Expansion

Time-resolved spectroscopy is a powerful tool for probing electron dynamics in molecules and solids, revealing transient phenomena on subfemtosecond time scales. The interpretation of experimental results is often enhanced by parallel numerical studies, which can provide insight and validation for experimental hypotheses. However, developing a theoretical framework for simulating time-resolved spectra remains a significant challenge. The most suitable approach involves the many-body nonequilibrium Green's function formalism, which accounts for crucial dynamical many-body correlations during time evolution. While these dynamical correlations are essential for observing emergent behavior in time-resolved spectra, they also render the formalism prohibitively expensive for large-scale simulations. Substantial effort has been devoted to reducing this computational cost─through approximations and numerical techniques─while preserving the key dynamical correlations. The ultimate goal is to enable first-principles simulations of time-dependent systems ranging from small molecules to large, periodic, multidimensional solids. Here, in this perspective, we outline key challenges in developing practical simulations for time-resolved spectroscopy, with a particular focus on Green's function methodologies. We highlight a recent advancement toward a scalable framework: the real-time Dyson expansion (RT-DE) [Phys. Rev. Lett. 2024, 133, 226902]. We introduce the theoretical foundation of RT-DE and discuss strategies for improving scalability, which have already enabled simulations of system sizes beyond the reach of previous fully dynamical approaches. We conclude with an outlook on future directions for extending RT-DE to first-principles studies of dynamically correlated, nonequilibrium systems.

Reeves, Cian C. [Univ. of California, Santa Barbar↗

DOE OSTI2026

Heterogeneous Multi-Domain Dataset Synthesis to Facilitate Privacy and Risk Assessments in Smart City IoT

The emergence of the Smart Cities paradigm and the rapid expansion and integration of Internet of Things (IoT) technologies within this context have created unprecedented opportunities for high-resolution behavioral analytics, urban optimization, and context-aware services. However, this same proliferation intensifies privacy risks, particularly those arising from cross-modal data linkage across heterogeneous sensing platforms. To address these challenges, this paper introduces a comprehensive, statistically grounded framework for generating synthetic, multimodal IoT datasets tailored to Smart City research. The framework produces behaviorally plausible synthetic data suitable for preliminary privacy risk assessment and as a benchmark for future re-identification studies, as well as for evaluating algorithms in mobility modeling, urban informatics, and privacy-enhancing technologies. As part of our approach, we formalize probabilistic methods for synthesizing three heterogeneous and operationally relevant data streams—cellular mobility traces, payment terminal transaction logs, and Smart Retail nutrition records—capturing the behaviors of a large number of synthetically generated urban residents over a 12-week period. The framework integrates spatially explicit merchant selection using K-Dimensional (KD)-tree nearest-neighbor algorithms, temporally correlated anchor-based mobility simulation reflective of daily urban rhythms, and dietary-constraint filtering to preserve ecological validity in consumption patterns. In total, the system generates approximately 116 million mobility pings, 5.4 million transactions, and 1.9 million itemized purchases, yielding a reproducible benchmark for evaluating multimodal analytics, privacy-preserving computation, and secure IoT data-sharing protocols. To show the validity of this dataset, the underlying distributions of these residents were successfully validated against reported distributions in published research. We present preliminary uniqueness and cross-modal linkage indicators; comprehensive re-identification benchmarking against specific attack algorithms is planned as future work. This framework can be easily adapted to various scenarios of interest in Smart Cities and other IoT applications. By aligning methodological rigor with the operational needs of Smart City ecosystems, this work fills critical gaps in synthetic data generation for privacy-sensitive domains, including intelligent transportation systems, urban health informatics, and next-generation digital commerce infrastructures.

IoT↗

NASA NTRSDate not supplied

The Effect of Temperature on the Preservation of Volatile-Rich Lunar Samples

Introduction. The Moon’s south pole is a high-priority target for human exploration and scientific study. This interest is, in part, due to the presence of Permanently Shadowed Regions (PSRs), which could contain high concentrations of unique volatiles at cryogenic temperatures [1]. Returned samples from PSRs may include a unique combination of rocks, regolith, and volatile species, providing unprecedented insights into the history of the Solar System and the potential for resource utilization on the Moon. However, because PSR samples are cryogenic up-on collection, lunar polar sample return will eventually require cold stowage for the journey from the Moon to Earth. Without cold stowage, PSR sample return will likely result in phase changes and chemical reactions within the volatile component of the sample, which would negatively impact the resulting scientific studies of those samples. This abstract summarizes the initial results from an ongoing characterization of analog PSR samples at a range of temperatures, with the goal of defining the temperatures needed for a flight cold stowage freezer. Background. Based on remote sensing observations of the Moon [2], south polar PSRs range in temperature from ~120K for small and/or shallow PSRs to ~20K at the most extreme locations in large, deep PSRs. A range of volatiles have been hypothesized to exist at the surface or subsurface of the lunar poles [3-5 and others]. This hypothesis was verified when the LCROSS mission impacted the <50-K PSR in the crater Cabeus, detecting a range of volatiles from water to low condensation temperature species such as H2S and methane [6]. Species such as H2S and ammonia (also detected by LCROSS) are also highly reactive, and increase the likelihood of chemical reactions at elevated (non-cryogenic) temperatures. At the Johnson Space Center’s Planetary Exploration and Astromaterials Research Laboratory (JSC-PEARL), we have developed a volatile-bearing lunar simulant that incorporates several of the species detected by LCROSS [Table 1] mixed cryogenically with the USGS Lunar Highlands Type (LHT) regolith simulant. The new volatile-regolith simulant will be used to assess the degree of sample alteration at room temperature, -20°C, -80°C, and -196°C (liquid nitro-gen), over a two-week period. Room temperature samples represent those likely to be returned during initial missions without cold stowage, -20°C provides an analog to Apollo cold curated samples, -80°C is the temperature of multiple flight payload freezers (e.g., MELFI), and -196°C is analogous to lunar PSRs. Two weeks is an approximation of the time between sample collection and Earth return for initial Artemis missions. Over this period of time, sample head-space gases will be analyzed using a Universal Gas Analyzer (UGA, a type of mass spectrometer) coupled with a Baratron pressure sensor. After testing, the regolith component of the simulant will be purged of volatiles and preserved for future electron beam and/or FTIR analysis. Experimental Procedure. Volatile-regolith simulants will be produced as an initial homogenous batch; this batch will then be distributed into aliquots (gas chromatography/GC vials or cryo vials), ensuring that each sample has the same starting composition and conditions [Fig. 1]. In addition to the “full” simulant shown in Table 1, less complex simulant compositions will be used as baseline and control samples [Table 2]. Aliquots will be produced in triplicate for each simulant composition, storage temperature, and date of sampling. Headspace gases in all Day 0 samples will be analyzed by the UGA immediately. Cold storage samples for future analytical days will be placed in freezers appropriate to their target temperatures (-20°C, -80°C, -196°C). For ambient-temperature samples, regolith and regolith-water samples will be stored in a fume hood, while the full simulant will be stored in a sealed Parr vessel for safety; no other simulants (RWCM/ RWCM+) will be stored at ambient temperature for this test. Samples will be analyzed by UGA in this manner on each Analysis Day outlined in Table 2. Analytical Data. The UGA measures the partial pressures in a single sample aliquot over a set mass range of 0-105 atomic mass units (AMU) [Figure 2]; this set range was selected to slightly exceed the mass of the highest-mass expected reaction product (H2SO4). The Baratron complements the UGA by measuring the total pressure in the headspace of a sample vial. Coupled together, the quantitative abundances of gases will be monitored throughout the test. UGA analyses of the triplicate samples for each storage temperature, day, and simulant composition will be averaged, and standard deviations for each will be calculated. The compositions of starting species (shown in Table 1) will be characterized as a function of time, and the presence of any new compounds (reaction products) will be monitored as well. Total pressures will be recorded for each sample analysis, and any samples that show signs of leakage (e.g., a significant reduction in pressure or simulant volatiles) will be discarded. Anticipated Results. Testing is planned to begin in January 2022. The resulting data will allow compositional and phase changes in the volatile component of the simulants to be determined. Both the reduction in initial compounds and the addition of reaction products are expected to be observed. In addition, the relative efficacy of the different temperatures at pre-serving the initial composition of the simulants will be quantified. Finally, the regolith component of each sample will be argon-purged and stored in a controlled environment for future laboratory analysis. Compositional and morphological changes in the regolith are expected for samples above 0°C. This test will be repeated three times over the course of 2022. Understanding the effect of temperature on both the volatile and regolith components of analog lunar materials will allow requirements for a cold stowage freezer to be developed. The implementation of cold stowage for lunar polar missions will maximize the preservation of returned samples, enabling ground-breaking lunar and Solar System volatiles science for decades to come.

J L Mitchell↗

NASA NTRS2015

PAHs in the Ices of Saturn's Satellites: Connections to the Solar Nebula and the Interstellar Medium

Aliphatic hydrocarbons and PAHs have been observed in the interstellar medium (e.g., Allamandola et al. 1985, Pendleton et al. 1994, Pendleton & Allamandola 2002, Tielens 2013, Kwok 2008, Chiar & Pendleton 2008) The inventory of organic material in the ISM was likely incorporated into the molecular cloud in which the solar nebula condensed, contributing to the feedstock for the formation of the Sun, major planets, and the smaller icy bodies in the region outside Neptune's orbit (transneptunian objects, or TNOs). Additional organic synthesis occurred in the solar nebula (Ciesla & Sandford 2012). Saturn's satellites Phoebe, Iapetus, and Hyperion open a window to the composition of one class of TNO as revealed by the near-infrared mapping spectrometer (VIMS) on the Cassini spacecraft at Saturn. Phoebe (mean diameter 213 km) is a former TNO now orbiting Saturn (Johnson & Lunine 2005). VIMS spectral maps of Phoebe's surface reveal a complex organic spectral signature consisting of prominent aromatic (CH) and aliphatic hydrocarbon (=CH2, -CH3) absorption bands (3.2-3.6 micrometers). Phoebe is the source of a huge debris ring encircling Saturn, and from which particles ((is) approximately 5-20 micrometers size) spiral inward toward Saturn (Verbiscer et al. 2009). They encounter Iapetus and Hyperion where they mix with and blanket the native H2O ice of those two bodies. Quantitative analysis of the hydrocarbon bands on Iapetus demonstrates that aromatic CH is approximately 10 times as abundant as aliphatic CH2+CH3, significantly exceeding the strength of the aromatic signature in interplanetary dust particles, comet particles, and in carbonaceous meteorites (Cruikshank et al. 2014). A similar excess of aromatics over aliphatics is seen in the qualitative analysis of Hyperion and Phoebe itself (Dalle Ore et al. 2012). The Iapetus aliphatic hydrocarbons show CH2/CH3 (is) approximately 4, which is larger than the value found in the diffuse ISM ((is) approximately 2-2.5). Insofar as Phoebe is a primitive body that formed in the outer regions of the solar nebula and has preserved some of the original nebula inventory, it can be key in understanding the content and degree of processing of that nebular material. A dynamical subset of TNOs define the Kuiper Belt, from which the short-period comets originate. Particles collected from comet 81P/Wild contain PAHs with an interstellar signature of deuterium. By inference, the PAHs contained in Phoebe and now dusted on the surfaces of two other Saturn satellites share that interstellar origin. There are other Phoebe-like TNOs that are presently beyond our ability to study in the organic spectral region, but JWST will open that possibility for a number of objects.

Solar nebula↗

DOE OSTI2021

Fourier Neural Networks as Function Approximators and Differential Equation Solvers

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a straightforward architecture with a single hidden layer. The simplicity of this network architecture facilitates the integration with any other higher-complexity networks, at a data pre- or postprocessing stage. We validate this FNN on naturally periodic smooth functions and on piecewise continuous periodic functions. We showcase the use of this FNN for modeling or solving partial differential equations with periodic boundary conditions. The main advantages of the current approach are the validity of the solution outside the training region, interpretability of the trained model, and simplicity of use.

Fourier decomposition↗