Bincike Kan Samfuran Lissafi na Analog

Teburin Abubuwan Ciki

1. Gabatarwa

Lissafin analog yana ba da fassara biyu a kimiyyar kwamfuta: lissafi ta hanyar kwatance da lissafi akan adadi masu ci gaba. A tarihi, an ƙera tsarin analog don yin ci gaba daidai da tsarin da suka tsara, yayin da fahimtar zamani ta jaddada yanayin ci gaba na lissafi sabanin lissafin lamba mai rabe-rabe.

Muhimman Hasashe

Lissafin analog yana haɗa lissafi mai ci gaba da ka'idar lissafi
Yawancin injunan analog na tarihi tsarin gauraye ne
Bambanci tsakanin rabe-rabe da ci gaba ba cikakke bane
Tsarin dynamical yana ba da tsarin haɗin kai

2. Tsarin Tsarin Dynamical

2.1 Tushen Lissafi

An ayyana tsarin dynamical a ka'ida a matsayin aikin ƙungiya mai ragi $T$ na $\mathbb{R}$ akan sarari $X$, wanda ke da siffa ta aikin kwarara $\phi: T \times X \rightarrow X$ wanda ya gamsar da:

$$\phi(0,x) = x$$

$$\phi(t, \phi(s,x)) = \phi(t+s,x)$$

2.2 Rarraba Lokaci

Ƙungiyoyin ragi na $\mathbb{R}$ ko dai suna da yawa a cikin $\mathbb{R}$ ko kuma suna da siffa iri ɗaya da lambobi, wanda ke haifar da tsarin lokaci mai ci gaba da tsarin lokaci rabe-rabe bi da bi.

3. Rarraba Samfura

3.1 Tasirin Sarari-Lokaci

Binciken ya gabatar da cikakkiyar rarrabuwar samfuran lissafi dangane da halayen lokaci da sarari:

Lokaci Mai Ci Gaba/Sarari Mai Ci Gaba

Cibiyoyin sadarwar jijiyoyi na analog, Daidaitattun lissafi

Lokaci Rabe-rabe/Sarari Mai Ci Gaba

Bincike mai maimaitawa, Samfurin BSS

Lokaci Mai Ci Gaba/Sarari Rabe-rabe

Yarjejeniyar jama'a, Cibiyoyin sadarwar sinadarai

3.2 Tsarin Gauraye

Yawancin tsarin analog na aiki suna nuna halayen gauraye, suna haɗa abubuwa masu ci gaba da rabe-rabe a cikin aikin su.

4. Tsarin Fasaha

4.1 Ƙirar Lissafi

Don tsarin da ke ci gaba da banbancewa, ana iya bayyana yanayin motsi kamar haka:

$$y' = f(y)$$

inda $f(y) = \frac{d}{dt}\phi(t,y)\big|_{t=0}$

4.2 Daidaiton Lissafi

Binciken ya kafa alaƙa tsakanin samfuran analog da ka'idar lissafi ta gargajiya, yana nuna cewa yawancin tsarin ci gaba na iya kwaikwayi injunan Turing da kuma akasin haka.

5. Sakamakon Gwaji

Takardar ta tattauna aiwatar da gwaji daban-daban na samfuran lissafi na analog, ciki har da:

Aiwatar da da'irar lantarki na masu warware daidaitattun lissafi
Cibiyoyin sadarwar sinadarai suna yin ayyukan ma'ana
Tsarin lissafi na gani don takamaiman ayyukan lissafi

Hoto na 1: Zanen Rarraba Samfuri

Zanen rarrabuwa yana kwatanta matsayi na samfuran lissafi daban-daban a cikin ci gaba na sarari-lokaci, yana nuna alaƙar da ke tsakanin lissafin lamba na gargajiya, tsarin analog, da hanyoyin gauraye masu tasowa.

6. Aiwar Code

A ƙasa akwai aiwar Python da ke nuna sauƙaƙan samfurin lissafi na analog ta amfani da daidaitattun lissafi na yau da kullun:

import numpy as np
from scipy.integrate import solve_ivp

class AnalogComputer:
    def __init__(self, system_function):
        self.f = system_function
    
    def compute(self, initial_conditions, time_span):
        """
        Warware tsarin dynamical: dy/dt = f(y)
        
        Parameters:
        initial_conditions: array-like, yanayin farko
        time_span: tuple (t_start, t_end)
        
        Returns:
        Magani object daga solve_ivp
        """
        solution = solve_ivp(
            self.f, 
            time_span, 
            initial_conditions,
            method='RK45'
        )
        return solution

# Misali: Tsarin layi
def linear_system(t, y):
    A = np.array([[-0.1, 2.0], [-2.0, -0.1]])
    return A @ y

# Fara da gudanar da lissafi
computer = AnalogComputer(linear_system)
result = computer.compute([1.0, 0.0], (0, 10))

7. Aikace-aikace da Hanyoyin Gaba

Samfuran lissafi na analog suna samun aikace-aikace a cikin:

Tsarin lissafi na Neuromorphic
Tsarin sarrafa lokaci-lokaci
Lissafin kimiyya da kwaikwayo
Na'urorin gefe da na'urorin IoT

Hanyoyin bincike na gaba sun haɗa da:

Gine-ginen analog-lamba gauraye
Lissafin analog mai wahayi na ƙidaya
Tsarin AI na analog masu ingantaccen makamashi
Tabbatar da ingancin tsarin analog

Bincike na Asali

Wannan binciken na Bournez da Pouly yana ba da cikakken tsarin don fahimtar lissafin analog ta hanyar hangen nesa na ka'idar tsarin dynamical. Marubutan sun yi nasarar haɗa ra'ayin tarihi na "lissafi ta hanyar kwatance" tare da tsarin lissafi na ci gaba na zamani, suna nuna cewa bambanci tsakanin lissafin analog da na lamba ya fi fahimta fiye da yadda ake sani.

Tushen lissafi da aka gabatar, musamman ma ƙirar tsarin dynamical ta amfani da ayyukan kwarara $\phi: T \times X \rightarrow X$, yana ba da ingantaccen tushe don nazarin kaddarorin lissafi na tsarin ci gaba. Wannan hanya ta yi daidai da ci gaban kwanan nan a cikin lissafin neuromorphic, inda tsarin kamar Loihi na Intel da guntu na TrueNorth na IBM suke aiwatar da ƙa'idodi masu kama da waɗanda aka tattauna a cikin binciken.

Musamman ma, rarrabuwar samfura dangane da halayen lokaci da sarari yana ba da hasashe masu mahimmanci don fahimtar iyawar lissafi na tsarin daban-daban. Haɗa samfuran da ba na al'ada ba kamar yarjejeniyar jama'a da cibiyoyin sadarwar sinadarai suna nuna faɗin lissafin analog fiye da na'urorin lantarki na analog na gargajiya.

Idan aka kwatanta da samfuran lissafi na lamba, tsarin analog suna ba da fa'idodi masu yuwuwa a cikin ingantaccen makamashi da yawan lissafi don takamaiman azuzuwan matsala, kamar yadda bincike daga cibiyoyi kamar ƙungiyar Analog VLSI da Signal Processing Group ta MIT ta tabbatar. Duk da haka, ƙalubale sun rage a cikin shirye-shirye, daidaito, da tabbatar da inganci, wuraren da tsarin lamba suka yi fice.

Ƙarfafa binciken akan tsarin gauraye yana nuna halin da ake ciki a cikin gine-ginen lissafi, inda tsarin kamar Tensor Processing Units (TPUs) na Google ke haɗa lissafi mai kama da analog don hasashe na cibiyar sadarwa yayin riƙe shirye-shiryen lamba. Wannan hanyar gauraye na iya wakiltar shirin gaba na tsarin lissafi na analog na aiki.

Nassoshi ga aikin tushe a ka'idar lissafi, kamar samfurin Blum-Shub-Smale (BSS) da bincike mai maimaitawa, suna ba da mahimmancin mahallin don fahimtar iyakokin ka'idar lissafin analog. Alaƙar da aka kafa tsakanin tsarin ci gaba da ka'idar lissafi ta gargajiya tana nuna cewa yawancin hasashe daga kimiyyar kwamfuta za a iya canjawa zuwa yankunan analog.

8. Nassoshi

Bournez, O., & Pouly, A. (2018). A Survey on Analog Models of Computation. arXiv:1805.05729
Blum, L., Shub, M., & Smale, S. (1989). On a theory of computation and complexity over the real numbers. Bulletin of the American Mathematical Society
Moore, C. (1990). Unpredictability and undecidability in dynamical systems. Physical Review Letters
Siegelmann, H. T., & Sontag, E. D. (1994). Analog computation via neural networks. Theoretical Computer Science
MIT Analog VLSI and Signal Processing Group. (2023). Recent Advances in Analog Computation
Intel Neuromorphic Computing Lab. (2022). Loihi 2: An Analog-Inspired Digital Architecture