Intent Field Theory of Biology: A Lagrangian Framework Linking Genome Architecture, Epigenetic Memory, and Phenotypic Dynamics

The emergence of biological complexity from genomic information remains one of the most profound challenges in modern science. While molecular biology has successfully cataloged the components of the cellular machinery, a unifying theoretical framework that describes the dynamical laws governing the integration of these components into a coherent phenotype is still missing. Historical models, such as Turing’s reaction-diffusion systems, provided early insights into pattern formation, but these often lacked a direct mathematical link to the structural constraints of the genome and the temporal persistence of epigenetic states.1 Intent Field Theory of Biology (IFTB) seeks to bridge this gap by proposing that biological systems are governed by a scalar field—the Intent field—defined over a biological spacetime. This field represents the integrated regulatory state of the organism, and its evolution is dictated by a Lagrangian density that accounts for genomic architecture, epigenetic memory, and environmental forcing. By framing biological decision-making as a variational problem, IFTB provides a rigorous mathematical structure that unifies developmental biology, genomics, and biophysics into a single predictive framework.2

Theoretical Foundations of the Intent Field

The conceptual genesis of IFTB lies in the observation that biological systems exhibit goal-directed behaviors, often termed "anatomical homeostasis" or "target morphology".2 These systems possess a remarkable ability to reach a specific anatomical endpoint despite perturbations, injury, or varying initial conditions. This suggests the existence of a set-point or an "informational template" that guides cellular collectives.2 In physics, such global coordination is typically described using field theories. IFTB formalizes this "informational template" as a scalar field , where represents coordinates in biological spacetime.

Biological spacetime is defined not merely by physical distance but by metabolic and developmental scales. The field encapsulates the instantaneous regulatory configuration, representing the "intent" of the biological unit—be it a single cell or a multicellular tissue—to move toward a specific phenotypic state.3 The dynamics of this field are derived from the principle of least action, a fundamental postulate in theoretical physics which states that the path taken by a system is the one that minimizes the action integral of the Lagrangian density .

The Lagrangian Formulation

The Lagrangian density for the Intent field is constructed to represent the energetic and informational costs associated with regulatory transitions. The standard form of the IFTB Lagrangian is expressed as:

The kinetic term represents the dynamic cost of changing the regulatory state. In a biological context, this corresponds to the metabolic energy required for chromatin remodeling, transcription factor translocation, and protein synthesis.1 The spatial gradient reflects the "tension" between neighboring cells, ensuring that tissues maintain spatial coherence. This spatial coupling is physically mediated by gap junctions, which allow for the synchronization of bioelectric potentials across cellular networks.2

Potential Landscape and Attractor States

The potential term defines the stability of various phenotypic states, serving as a quantitative representation of the Waddington epigenetic landscape.7 To capture the bistability observed in cellular differentiation—where a progenitor can choose between two distinct lineages—a double-well potential is employed:

In this potential, represents the stable attractor states, corresponding to fully differentiated cell types or stable behavioral modes.9 The parameter determines the height of the "chreodes" or ridges that separate these valleys, representing the degree of canalization.7 High values of imply a highly canalized system where the phenotype is robust to noise, whereas lower values suggest a plastic state susceptible to reprogramming.8

Genomic Architecture as a Structural Constraint

A unique feature of IFTB is the explicit incorporation of genomic architecture into the field dynamics. DNA is not merely an abstract sequence of bits but a physical polymer with heterogeneous mechanical properties that dictate its accessibility and regulatory potential.11

Mechanical Properties of DNA and Field Coupling

The sequence of DNA significantly influences its secondary structure and flexibility. High-throughput cyclizability assays, such as Loop-seq, have demonstrated that the "mechanical code" of DNA is as critical as its primary sequence.12 GC-rich regions exhibit higher stiffness and a larger stretch modulus compared to AT-rich regions, which are more "loopable" and flexible.11 In the IFTB Lagrangian, this mechanical heterogeneity is represented by the genomic coupling term:

Here, represents the local density of GC base pairs. The coupling constant acts as a "mass" term for the Intent field, where high GC content increases the effective inertia of the field, anchoring it to specific regulatory states.11 Conversely, the flexible AT-rich regions, often found in TATA boxes and flexible linkers, allow for rapid field reconfigurations, facilitating the initiation of transcription and other dynamic regulatory events.12

Genomic Feature	Biophysical Property	IFTB Role	Source ID
GC Content	High stiffness, larger stretch modulus	Field Anchor ()	11
AT Content	High flexibility, high cyclizability	Field Plasticity	12
CpG Dinucleotides	Subject to methylation	Hysteresis Modulator ()	12
Minor Groove Width	Electrostatic potential source	Local Field Charge ()	14

Genomic Source Terms and Language Model Embeddings

The genome also provides active source terms that "drive" the Intent field toward specific configurations. These source terms represent the presence of promoters, enhancers, and other regulatory elements. Modern genome language models (gLMs), such as Nucleotide Transformer and DNABERT, can be used to extract these functional features.15 By processing raw DNA sequences, these models generate high-dimensional embeddings that encapsulate the regulatory potential of a genomic region.17 In IFTB, these embeddings are mapped to the spatial distribution of the source term , effectively turning the genome into a "charge distribution" that shapes the Intent field.18

Epigenetic Memory and Transgenerational Dynamics

Epigenetic states—such as DNA methylation and histone modifications—provide a mechanism for the system to store its history, creating a form of memory that influences future field dynamics.20 IFTB models this through the term , where is the epigenetic state.

Hysteresis and Cellular Memory

The interaction between the Intent field and the epigenetic state generates hysteresis. When a biological system is exposed to an environmental stimulus , the Intent field shifts. This shift, in turn, alters the epigenetic state through feedback mechanisms like the "self-sustaining feedback loops" described in developmental literature.20 Once the stimulus is removed, the modified prevents the field from returning to its original state immediately, creating a lasting "memory" of the event.13

Transgenerational Inheritance Laws

The transgenerational transmission of these memory states has been rigorously characterized in model organisms like C. elegans. Research by the Rechavi group has identified three fundamental "laws" that govern the inheritance of small RNA-mediated silencing 24:

1. Uniformity of Distribution: Silencing initiated by a mother is distributed evenly among all descendants in a given generation.24

2. Stochastic Inheritance States: Isogenic individuals assume different "inheritance states" stochastically, determining the duration of the transgenerational response.24

3. Positive Feedback of Persistence: The likelihood of continued inheritance increases with the number of generations the response has already persisted.24

In IFTB, these laws serve as the evolution equations for the epigenetic field . The third law, in particular, suggests a "reinforcement" mechanism where the coupling becomes stronger the longer the system remains in a particular attractor basin, mirroring the way habitual behaviors or chronic disease states become "locked in" over time.20

Equation of Motion: The Nonlinear Klein-Gordon Framework

Applying the Euler-Lagrange equation to the IFTB Lagrangian yields a nonlinear, second-order partial differential equation that describes the spatio-temporal evolution of biological intent.

Derivation and Physical Significance

The resulting equation of motion is:

where is the wave operator, including a diffusion constant to account for the passive spread of regulatory signals.26 This equation is a variant of the nonlinear Klein-Gordon equation, a fundamental model in relativistic quantum mechanics and condensed matter physics.29

The physical interpretation of this equation in biology is profound. It suggests that biological "intent" behaves as a wave propagating through a structured medium (the genome and the epigenetic state). The solutions to this equation include solitons—stable, self-reinforcing wave packets that can travel across tissues without dispersing.26 These solitons represent coherent biological structures or "decision units" that maintain their identity despite the turnover of individual cells.1

Attractor Stability and Disease States

The stability of the Intent field depends on the balance between the internal potential and the external couplings. In healthy development, the field is trapped in a stable attractor well . However, if the coupling to the genome () or epigenetic memory () is weakened—as seen in aging or oncogenesis—the field may escape its basin and enter an unstable regime.1 This "field instability" model of cancer suggests that tumors are not merely collections of mutated cells but are regions where the Intent field has lost its coherence, leading to the breakdown of anatomical homeostasis.1

Computational Simulation using Physics-Informed Neural Operators

Because the IFTB equation is nonlinear and depends on complex, multi-dimensional genomic inputs, direct analytical solutions are generally unavailable. A computational framework is required that can simulate the Intent field across diverse virtual populations.

Genome Embedding and Parameter Mapping

The simulation begins with the encoding of whole-genome sequences using transformer-based gLMs.16 Models like GENERator-v2 or Nucleotide Transformer process segments of DNA (up to 98k base pairs) and generate vectors that capture the "regulatory grammar" of the sequence.17 These embeddings are then used to define the spatial parameters of the PDE:

1. Spatial Distribution of Source : Mapped from the predicted locations of enhancers and promoters.15

2. Genomic Coupling : Mapped from local base composition and structural motifs.12

3. Potential Barrier : Derived from the density of canalizing motifs.7

The PINO Architecture

To solve the Intent field PDE efficiently, IFTB utilizes Physics-Informed Neural Operators (PINOs). PINOs represent a hybrid approach that combines the data-driven power of neural operators with the physical consistency of PINNs.35 Unlike traditional numerical solvers (like Finite Element Methods) that must be re-run for every new genomic configuration, a PINO learns the operator that maps genomic inputs to the resulting field trajectory .36

The loss function for training the PINO includes a data loss term (to match experimental observations) and a physics loss term (the residual of the Klein-Gordon equation):

This framework allows for real-time inference, enabling the simulation of thousands of individuals in a "virtual population" to study the effects of rare mutations or varying environmental histories.28

Quantitative Predictions and Experimental Falsifiability

A key requirement of IFTB is that it must generate quantitative, testable predictions that could, in principle, falsify the theory.

Prediction 1: Correlation between GC-Content and Regulatory Inertia

IFTB predicts that the rate at which a gene can change its expression level in response to a stimulus is inversely proportional to the GC content of its regulatory domain. Specifically, the genomic coupling constant should be a linear function of the local stretch modulus measured by biophysical assays.11 In single-cell transcriptomic experiments, this would manifest as "regulatory inertia," where GC-rich genes show slower response times but higher stability compared to AT-rich genes.12

Prediction 2: Morphological vs. Behavioral CRISPR Outcomes

The theory distinguishes between the spatial gradient term () and the potential well term (). It predicts that CRISPR-mediated edits in AT-rich "flexible" regions (which govern the gradient) will primarily lead to morphological defects—such as altered organ shape—without changing the underlying behavioral choices of the system. Conversely, edits in GC-rich "stiff" regulatory islands will alter behavioral attractor states—switching the cell's fate or the organism's habit—without necessarily affecting its gross morphology.2

Prediction 3: Hysteresis Area as a Measure of Epigenetic Load

The theory provides a mathematical relationship between the area of a phenotypic hysteresis loop and the amount of epigenetic "load" in the system. If an organism is exposed to an environmental shift and then returned to its original state, the "lag" in its recovery (the hysteresis) should be directly proportional to the concentration of inherited small RNAs or methylation marks .13

Technical Implementation and Statistical Analysis

The implementation of the IFTB framework requires a multi-stage computational pipeline, integrating genomic data processing with nonlinear PDE optimization.

Data Acquisition and Preprocessing

Genomic sequences for the target species (e.g., C. elegans, D. melanogaster) are obtained from standard repositories like RefSeq. These sequences are tokenized into k-mers (e.g., 6-mers) and processed by the Nucleotide Transformer to generate embeddings , where is typically 768 or 1024.17 These embeddings are then mapped to the spatial grid of the biological domain using a linear projection layer trained on known regulatory annotations.15

Stage	Algorithm/Tool	Output	Source ID
Sequence Embedding	Nucleotide Transformer (v2)	Feature Vectors ()	17
Parameter Mapping	Differentiable Linear Layer		18
PDE Integration	Physics-Informed Neural Operator	Intent Field	36
Phenotype Mapping	MLP with Sigmoid Output	Phenotypic State	40

Parameter Identification and Inverse Modeling

To fit the theory to experimental data, an inverse modeling approach is used. Given a set of observed phenotypes under different conditions , the parameters of the Lagrangian are identified by minimizing the joint loss:

This optimization is performed using the Adam optimizer followed by L-BFGS to ensure convergence to a high-precision local minimum.3 The use of automatic differentiation allows for the calculation of gradients with respect to the genomic coupling constants, effectively "mapping" the phenotypic variation back to specific genomic loci.40

Statistical Power and Virtual Populations

The robustness of the theory is assessed through "virtual population" studies. By simulating individuals with randomized genomes and environmental histories, the PINO solver can generate a distribution of phenotypes. This allow for the calculation of "field-theoretic heritability," a new metric that quantifies the proportion of phenotypic variance explained by the dynamics of the Intent field versus stochastic noise.24

Nature-Style Figures for IFTB

Figure 1: The Lagrangian Architecture of Biological Intent

This figure serves as the conceptual anchor of the paper, illustrating the hierarchical integration of molecular and field-theoretic layers.

• Panel (a): The Genomic-Mechanical Interface. A high-resolution schematic of DNA as a physical polymer. Regions of high GC density are rendered as rigid, high-inertia segments, while AT-rich regions are shown as flexible, low-inertia loops. This panel visually represents the coupling constant as a structural parameter of the "medium" in which the Intent field propagates.

• Panel (b): The Potential Surface . A 3D landscape visualization of the double-well potential. Progenitor cells are depicted as "marbles" on a ridge, with arrows showing trajectories toward differentiated "valleys" (). The height of the ridge is explicitly labeled as the canalization parameter .

• Panel (c): Interaction of Field and Environment. A diagram showing how external environmental signals and epigenetic memory tilt the potential landscape, shifting the location of the attractors and driving the Intent field wave across the tissue.

Figure 2: PINO-Based Simulation and Virtual Populations

This figure demonstrates the computational power of the framework.

• Panel (a): The PINO Architecture. A technical flowchart showing the input (Genomic Embedding ), the operator network (Fourier integral kernels), and the output (Spatio-temporal field ). The physics loss term is highlighted as a constraint that ensures the output satisfies the nonlinear Klein-Gordon equation.

• Panel (b): Attractor Trajectories. Plots of field intensity over time for multiple individuals. Stable phenotypes appear as horizontal lines at , while developmental transitions appear as rapid "jumps" or phase transitions between wells.

• Panel (c): Population Phenotypic Distribution. A heatmap showing the range of morphological outcomes predicted for a population with varying genomic values. This illustrates the theory's ability to predict population-level variance from first principles.

Figure 3: Transgenerational Memory and Hysteresis Loops

This figure explores the temporal persistence of the Intent field.

• Panel (a): Pathogen Avoidance in C. elegans. A schematic of the Coleen Murphy experiment, where exposure to P11 small RNA triggers avoidance.24

• Panel (b): The Hysteresis Curve. A plot of avoidance behavior () against pathogen exposure (). The forward and return paths are distinct, with the gap between them representing the "epigenetic load" .

• Panel (c): Multi-Generational Decay. A series of plots showing the hysteresis loop gradually shrinking across generations F1 to F4, corresponding to the "dissipation" of the heritable RNAi response according to the Rechavi laws.25

Figure 4: Field Instability and the "Cancer Landscape"

This figure applies IFTB to pathology.

• Panel (a): The Breakdown of Canalization. A comparison of a healthy potential landscape (deep wells, high ) and a "cancerous" landscape (flat, unstable surface).

• Panel (b): Soliton Disintegration. A simulation result showing how a stable "soliton" (a healthy tissue state) breaks down into chaotic, non-coherent oscillations when the genomic coupling is perturbed.

• Panel (c): Predictive Reprogramming. A demonstration of how applying a targeted "bioelectric field" can force the Intent field back into a stable attractor basin, providing a theoretical basis for regenerative therapies.

Mathematical Supplement: Detailed Derivations

Derivation 1: The Energy-Momentum Tensor

To understand the conservation laws in IFTB, we derive the energy-momentum tensor for the Intent field. The tensor is given by:

The component represents the "regulatory energy density" of the biological system:

This energy density must be non-negative for a stable biological state. A "negative energy" region would correspond to a runaway instability, potentially modeling the rapid, uncontrolled growth seen in aggressive malignancies.1

Derivation 2: The Non-Relativistic Limit

In biological systems where transitions are slow relative to the metabolic "speed of life," we can take the non-relativistic limit of the Klein-Gordon equation. By substituting , we recover a nonlinear Schrödinger-type equation:

This identifies biological morphogenesis as a process of "coherence maintenance," where the system behaves like a macroscopic quantum fluid or a "cosmic superfluid" at the biological scale.1

Derivation 3: Stability Analysis of the Attractor

We perform a linear stability analysis of the attractor by considering a small perturbation . The linearized equation for the perturbation is:

For stability, the term in the brackets must be positive. This leads to a quantitative "viability criterion" for biological systems:

This inequality defines the "viability envelope" of the organism. If mutations reduce or environment shifts such that this criterion is violated, the system undergoes a catastrophic collapse or "blow-up," as described in the Cauchy problem for the Klein-Gordon equation.32

Discussion and Evolutionary Implications

The Intent Field Theory of Biology shifts the paradigm from a gene-centric view of life to a field-centric view. In this framework, the genome is not a "blueprint" but a "refractive medium" that shapes the information field of the organism.

Evolution as the Optimization of the Lagrangian

Under IFTB, evolution can be viewed as the process of optimizing the parameters of the Lagrangian——to maximize the stability and robustness of the Intent field across diverse environments. Natural selection acts on the "topography" of the potential landscape, favoring genomic architectures that produce deep, well-canalized attractor wells.7 The "evolutionary role of bioelectricity" then becomes clear: it is a high-speed modulator of the field that allows for rapid adaptation without requiring genomic mutations.4

Philosophical and Scientific Synthesis

This theory erases the arbitrary boundary between the "computational" processes of the brain and the "morphogenetic" processes of the body. Both are expressions of the same Intent field dynamics, differing only in their time scales and coupling constants.2 By providing a single mathematical language for genomics, bioelectricity, and epigenetics, IFTB offers a path toward a truly "unified" biology—one that is consistent with the laws of physics yet captures the unique, goal-directed essence of life.1

The computational framework combining gLMs and PINOs provides the necessary bridge to apply this theory to real-world genomic data. As our ability to read and write genomes improves, the Intent Field Theory of Biology may provide the necessary "operating system" for the rational design of new biological forms and the precise correction of diseased states.2

Conclusion

The Intent Field Theory of Biology provides a robust, Lagrangian-based framework for understanding the complex dynamics of life. By representing biological decision-making as a scalar field evolving within a potential landscape shaped by genomic architecture and epigenetic history, the theory unifies disparate fields of biological inquiry into a single dynamical system. The proposed equation of motion—a nonlinear Klein-Gordon equation—captures the essential features of morphogenesis, attractor stability, and transgenerational memory. Coupled with a modern computational pipeline using physics-informed neural operators and genome language models, this framework offers a powerful new paradigm for predicting phenotypic outcomes and understanding the fundamental nature of biological intent. If experimentally validated, IFTB could revolutionize our approach to genetic engineering, regenerative medicine, and the study of evolutionary biology, marking the transition from a descriptive science to a predictive, field-theoretic discipline.

ผลงานที่อ้างอิง

1. Bioelectric Fields at the Beginnings of Life - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC9810354/

2. The bioelectric code: An ancient computational medium for dynamic ..., เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC10464596/

3. The Human as Electromagnetic Being: A Quaternion Biofield Model Integrating Bioelectricity, Personality Dynamics, and Energetic Structure, เข้าถึงเมื่อ มีนาคม 13, 2026 https://constable.blog/wp-content/uploads/The-Human-as-Electromagnetic-Being-.pdf

4. Bioelectric mechanisms in regeneration: unique aspects and future perspectives - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC2706303/

5. Novel Roles of Standard Lagrangians in Population Dynamics Modeling and Their Ecological Implications - MDPI, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.mdpi.com/2227-7390/11/17/3653

6. Bioelectricity in Morphogenesis - Annual Reviews, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.annualreviews.org/content/journals/10.1146/annurev-cellbio-101323-032747?crawler=true&mimetype=application/pdf

7. Conrad Waddington and his epigenetic landscape - Bitesize Bio, เข้าถึงเมื่อ มีนาคม 13, 2026 https://bitesizebio.com/10999/conrad-waddington-and-his-epigenetic-landscape/

8. A deterministic map of Waddington's epigenetic landscape for cell fate specification - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC3213676/

9. Quantifying the Waddington landscape and biological paths for development and differentiation - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC3100956/

10. Compelled by the Diagram: Thinking through C. H. Waddington's Epigenetic Landscape, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/281544682_Compelled_by_the_Diagram_Thinking_through_C_H_Waddington's_Epigenetic_Landscape

11. Mechanical properties of high-G.C content DNA with a-type base-stacking - PubMed, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pubmed.ncbi.nlm.nih.gov/21504736/

12. Deciphering the mechanical code of genome and epigenome | bioRxiv, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.biorxiv.org/content/10.1101/2020.08.22.262352.full

13. GC-Content Dependence of Elastic and Overstretching Properties of DNA:RNA Hybrid Duplexes - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC7451903/

14. Genome-wide prediction of minor-groove electrostatic potential enables biophysical modeling of protein–DNA binding | Nucleic Acids Research | Oxford Academic, เข้าถึงเมื่อ มีนาคม 13, 2026 https://academic.oup.com/nar/article/45/21/12565/4430925

15. Explainability in transformer models for functional genomics - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC8425421/

16. Transformers and genome language models - Moses Lab - University of Toronto, เข้าถึงเมื่อ มีนาคม 13, 2026 http://www.moseslab.csb.utoronto.ca/Consens_Wang_Nat_Mach_Int_2025.pdf

17. Functional In-Context Learning in Genomic Language Models with ..., เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.biorxiv.org/content/10.64898/2026.01.27.702015v1.full-text

18. Systems mapping: how to improve the genetic mapping of complex traits through design principles of biological systems - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC3127792/

19. Embed-Search-Align: DNA sequence alignment using Transformer models - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC11919449/

20. Transgenerational Epigenetic Inheritance: myths and mechanisms - PMC - NIH, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC4020004/

21. Transgenerational epigenetic inheritance | Greer Lab | Washington University in St. Louis, เข้าถึงเมื่อ มีนาคม 13, 2026 https://greerlab.wustl.edu/items/transgenerational-epigenetic-inheritance/

22. Transgenerational Epigenetic Inheritance: From Phenomena to Molecular Mechanisms - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC6889819/

23. Transgenerational epigenetic inheritance - Wikipedia, เข้าถึงเมื่อ มีนาคม 13, 2026 https://en.wikipedia.org/wiki/Transgenerational_epigenetic_inheritance

24. A Worm's-eye View of Transgenerational Epigenetic Inheritance, เข้าถึงเมื่อ มีนาคม 13, 2026 https://epigenie.com/c-elegans-studies-worm-into-the-fundamentals-of-transgenerational-epigenetic-inheritance/

25. Three Rules Explain Transgenerational Small RNA Inheritance in C ..., เข้าถึงเมื่อ มีนาคม 13, 2026 https://pubmed.ncbi.nlm.nih.gov/32841602/

26. (PDF) A variational iteration method for solving the nonlinear Klein-Gordon equation, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/286044008_A_variational_iteration_method_for_solving_the_nonlinear_Klein-Gordon_equation

27. Applications of the nonlinear Klein/Sinh-Gordon equations in modern physics: A numerical study - ResearchGate, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/383986019_Applications_of_the_nonlinear_KleinSinh-Gordon_equations_in_modern_physics_A_numerical_study

28. Physics-Informed Machine Learning in Biomedical Science and Engineering - arXiv.org, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/html/2510.05433v1

29. [2509.23191] On the rotating nonlinear Klein-Gordon equation with multiscale effects: structure-preserving methods and applications to vortex dynamics - arXiv, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/abs/2509.23191

30. Applications of the nonlinear Klein/Sinh-Gordon equations in modern physics: a numerical study - AIMS Press, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.aimspress.com/article/doi/10.3934/mmc.2024029?viewType=HTML

31. [1510.00415] Dissipative Effects in Nonlinear Klein-Gordon Dynamics - arXiv, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/abs/1510.00415

32. 'Relativistic' propagation of instability fronts in nonlinear Klein-Gordon equation dynamics, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/html/2603.09343v1

33. Stable and unstable sets for the Cauchy problem for nonlinear Klein-Gordon equation with potential - ResearchGate, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/267657242_Stable_and_unstable_sets_for_the_Cauchy_problem_for_nonlinear_Klein-Gordon_equation_with_potential

34. Stability of Breathers for a Periodic Klein–Gordon Equation - PMC, เข้าถึงเมื่อ มีนาคม 13, 2026 https://pmc.ncbi.nlm.nih.gov/articles/PMC11430978/

35. Applications of physics informed neural operators | Argonne National Laboratory, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.anl.gov/argonne-scientific-publications/pub/182976

36. Physics-Informed Fine-Tuned Neural Operator for Flow Field Modeling - MDPI, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.mdpi.com/2077-1312/14/2/201

37. Physics-Informed Neural Operator for Learning Partial Differential Equations - ICML 2026, เข้าถึงเมื่อ มีนาคม 13, 2026 https://icml.cc/virtual/2022/18719

38. (PDF) Physics-Informed Neural Networks and Neural Operators for Parametric PDEs: A Human-AI Collaborative Analysis - ResearchGate, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/397366383_Physics-Informed_Neural_Networks_and_Neural_Operators_for_Parametric_PDEs_A_Human-AI_Collaborative_Analysis

39. [2511.04576] Physics-Informed Neural Networks and Neural Operators for Parametric PDEs - arXiv, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/abs/2511.04576

40. Using Physics-Informed Neural Networks for Modeling Biological and Epidemiological Dynamical Systems - MDPI, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.mdpi.com/2227-7390/13/10/1664

41. EquiNO: A physics-informed neural operator for multiscale simulations - arXiv.org, เข้าถึงเมื่อ มีนาคม 13, 2026 https://arxiv.org/html/2504.07976v1

42. Understanding the Natural Language of DNA using Encoder-Decoder Foundation Models with Byte-level Precision - NSF PAR, เข้าถึงเมื่อ มีนาคม 13, 2026 https://par.nsf.gov/servlets/purl/10544572

43. Time Scale Gradient Aggregation Strategy for Systems Biology Parameter Identification - ResearchGate, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/397633677_Time_Scale_Gradient_Aggregation_Strategy_for_Systems_Biology_Parameter_Identification

44. (PDF) The Role of Bioelectrical Patterns in Regulative Morphogenesis: An Evolutionary Simulation and Validation in Planarian Regeneration - ResearchGate, เข้าถึงเมื่อ มีนาคม 13, 2026 https://www.researchgate.net/publication/392258427_The_Role_of_Bioelectrical_Patterns_in_Regulative_Morphogenesis_an_Evolutionary_Simulation_and_Validation_in_Planarian_Regeneration

45. Research of Ying Nian Wu - UCLA Statistics & Data Science, เข้าถึงเมื่อ มีนาคม 13, 2026 http://www.stat.ucla.edu/~ywu/research.html