浼楁墍鍛ㄧ煡锛岄噾铻嶅競鍦轰腑鐨勬椂闂村簭鍒楁暟鎹潅涔辨棤绔犱笖闅句互澶勭悊銆傝繖灏辨槸浜轰滑瀵归噾铻嶆暟瀛﹂鍩熷姝ゆ劅鍏磋叮鐨勯儴鍒嗗師鍥狅紒鎴戜滑鍙互鐢ㄦ潵鏇村ソ鍦扮悊瑙ｈ秼鍔匡紙鎴栧府鍔╂ā寮忚瘑鍒?棰勬祴绠楁硶锛夌殑涓€绉嶆柟娉曟槸鏃堕棿搴忓垪骞虫粦銆備互涓嬩紶缁熸柟娉曪細绉诲姩骞冲潎-绠€鍗曘€佸鏄撱€佹湁鏁堬紙浣嗙粰鏃堕棿搴忓垪鏁版嵁涓€涓瀵熺殑鈥滄粸鍚庘€濓級锛孲avitzky-Golay杩囨护鍣?鏈夋晥浣嗘洿澶嶆潅锛屽畠鍖呭惈涓€浜涚洿瑙傜殑瓒呭弬鏁扮殑浼犵粺鏂规硶鏄痟eatsolutionequation锛岃秴鍙傛暟鏇寸洿瑙傦紝閫熷害涔熷緢蹇紒鍦ㄦ湰鏂囦腑锛屾垜浠皢鑰冭檻涓€涓◢寰鏉傜殑鏂圭▼锛屼絾瀹冨叿鏈変繚瀛樿竟鐨勬晥鏋溿€傝繖涓柟绋嬬О涓篜erona-MalikPDE锛堝亸寰垎鏂圭▼锛夛紝鍏跺钩婊戞晥鏋滃彲浠ュ湪涓嬮潰鐨勫姩鐢讳腑鐪嬪埌锛氫笂鍥炬槸鐗规柉鎷夛紙TSLA锛夋敹鐩樹环鍦?022骞翠环鏍兼晥鏋滀腑浣跨敤鐨勮竟缂樺钩婊戞柟娉曘€傛爣棰樹腑鐨勨€渢=x鈥濆搴斾簬鎴戜滑瀵瑰簭鍒楄繘琛屽钩婊戝鐞嗙殑鏃堕棿锛堜互鏃犻噺绾插崟浣嶈〃绀猴級銆傚鏋滄偍瀵逛互涓婃晥鏋滄劅鍏磋叮锛岄偅涔堟湰鏂囧皢瑙ｉ噴浠ヤ笅鍐呭锛歅erona-MalikPDE锛堝亸寰垎鏂圭▼锛夛紝浠ュ強涓轰粈涔堜娇鐢ㄥ畠浠ュ強濡備綍姹傝В鍋忓井鍒嗘柟绋嬨€備笌鐑柟绋婸erona-MalikPDE鐨勬瘮杈冧笅闈㈡槸灏嗚澶勭悊鐨勬柟绋嬪叕寮忥細Perona-MalikPDE銆傚紡涓紝u鏄垜浠骞虫粦鐨勬椂闂村簭鍒楋紝伪鏄帶鍒惰竟缂樹繚鎶ょ殑鍙傛暟锛埼辫秺灏忓搴旇竟缂樹繚鎶よ秺澶氾級銆傜湅璧锋潵鏈夌偣澶嶆潅锛屼笅闈㈢户缁鏄庛€傚綋鎴戜滑灏濊瘯浠ュ師濮嬪舰寮忔眰瑙ｆPDE鏃讹紝瀹冧細瀵艰嚧涓€浜涢棶棰橈紱鎵€浠ユ垜浠€冭檻涓€绉嶄慨鏀瑰舰寮忥細鍩烘湰涓婏紝鍑芥暟g鍦ㄥ唴閮ㄦ墽琛岄珮鏂嵎绉紙鍗冲彉寰楁洿骞虫粦锛夈€傝繖绉颁负姝ｅ垯鍖栵紝鎴戜滑鍙渶瑕佺煡閬撳畠鏄彲瑙ｇ殑鍗冲彲銆傝繖涓彲鎬曠殑鏂圭▼寮忔瘮涓婇潰鐨勬洿澶嶆潅锛屼絾鏄繖閲屾垜浠病鏈夊涓┖闂寸淮搴︼紝鎴戜滑鍦ㄥ钩婊戜竴涓椂闂村簭鍒楋紝鎵€浠ュ畠鍙湁涓€涓淮搴︺€傛墍浠ラ渶瑕佽В鐨勬柟绋嬫槸锛氳繖鏍风湅璧锋潵绠€鍗曞浜嗭紝鑷冲皯鎴戜滑鐭ラ亾鎵€鏈夌殑瀛楁瘝馃榿锛屼絾鏄鏋滀綘鎯充簡瑙ｇ粏鑺傦紝浣犲彲浠ラ€氳繃瀵逛笂闈㈢殑鍏紡杩涜鎵╁睍鍜岀畝鍖栨潵鎺ㄥ寰楀埌锛岃櫧鐒朵笉寤鸿杩欐牱鍋氾紝浣嗗鏋滀綘鎰挎剰锛岃闅忔剰銆傚垰鎵嶆垜浠彁鍒版垜浠鐞嗙殑鏃堕棿搴忓垪鏄竴缁寸殑锛屼絾鏄负浠€涔堝亸寰垎鏂圭▼鏄簩缁寸殑鍛紵杩欎釜鍋忓井鍒嗘柟绋嬫槸鏍规嵁鏃堕棿姹傝В鐨勩€傚熀鏈笂锛屾瘡涓€姝ラ兘浼氳繘涓€姝ュ钩婊戞暟鎹€傛墍浠瓒婂ぇ锛屾椂闂村簭鍒楄秺骞虫粦锛屼篃灏辨槸璇寸┖闂村彉閲弜浠ｈ〃浜嗘椂闂村簭鍒椾腑鐨勨€滄椂闂粹€濓紝鍚庨潰浼氳缁嗚В閲婅В娉曘€備负浠€涔堣繖涓瓑寮忥紵鐑柟绋嬬殑闂鍦ㄤ簬瀹冧笉鑳藉緢濂藉湴淇濈暀杈圭紭銆備繚鐣欒繖浜涜竟缂樹互鎹曟崏蹇€熺殑浠锋牸澶у彉鍔ㄥ彲鑳芥槸鏄庢櫤鐨勶紝浣嗚鍘婚櫎浠讳綍灏忕殑浣嗛珮棰戠殑鍣煶銆傝繖绉嶆柟娉曟瘮鐑柟绋嬫洿闅撅紝鍥犱负Perona-MalikPDE鏄潪绾挎€х殑锛堜笌绾挎€х殑鐑柟绋嬩笉鍚岋級銆備竴鑸潵璇达紝闈炵嚎鎬ф柟绋嬩笉鍍忕嚎鎬ф柟绋嬮偅鏍峰鏄撴眰瑙ｃ€傚浣曡В鍐宠繖涓亸寰垎鏂圭▼锛屾垜浠皢浣跨敤涓€绉嶇О涓烘湁闄愬樊鍒嗙殑鏂规硶銆傚畠鏄竴绉嶆眰鍋忓井鍒嗭紙鎴栧父寰垎锛夋柟绋嬪拰鏂圭▼鐨勬暟鍊艰В鐨勬柟娉曘€傛偍鍙互灏嗗叾瑙嗕负姣忔閬囧埌涓嬪浘涓殑浜ゅ弶鐐规椂瀵绘壘瑙ｅ喅鏂规锛氫竴涓湪鏃堕棿鍜岀┖闂翠笂鍒嗚鎴愮鏁ｅ尯闂寸殑鍥俱€傝繖閲岀┖闂翠笂鐨勭鏁ｅ尯闂存槸浠嶽0,1]寮€濮嬬殑锛屾椂闂翠笂鐨勭鏁ｅ尯闂存槸浠巘=0鍒皌=sk锛屽叾涓璼鏄垜浠緱鍒扮殑鍖洪棿銆傜嚎鐨勪氦鐐规槸鎴戜滑鎵惧埌鍋忓井鍒嗘柟绋嬭В鐨勫湴鏂广€傚湪澶勭悊鏁板瓧涔嬪墠锛屾垜浠渶瑕佷粠鏁板涓婂畾涔夋暣涓棶棰樸€傜敱浜庢柟绋嬪湪绌洪棿涓婃槸浜岄樁鐨勶紝鍦ㄦ椂闂翠笂鏄竴闃剁殑锛屽洜姝ら渶瑕佷袱涓竟鐣屾潯浠跺拰涓€涓垵濮嬫潯浠讹細鎴戜滑灏嗘眰瑙ｆ柟绋嬬粍浠ュ钩婊戞椂闂村簭鍒楋紙杩欎釜鏂圭▼鐪嬭捣鏉ユ瘮浠ｇ爜澶嶆潅寰楀锛侊級锛屾垜浠殑寮€濮媝oint鏄偂浠锋椂闂村簭鍒楋紝缁堢偣鎬绘槸鐩稿悓鐨勪环鏍笺€傞偅涔堟垜浠浣曞紑濮嬫暟鍊兼眰瑙ｅ憿锛熸垜浠渶鍒濈殑鏂规硶鏄娇鐢ㄨ繖浜涘鏁扮殑鏈夐檺宸垎杩戜技锛屽嵆Perona-MalikPDE涓鏁扮殑杩戜技銆傝繖浜涘鏁扮殑鎺ㄥ涓嶅湪鏈枃璁ㄨ鑼冨洿鍐咃紝灏变笉璇︾粏鍐欎簡銆備笂寮忎腑锛宧鍜宬鍒嗗埆鏄┖闂村拰鏃堕棿涓婄鏁ｇ偣涔嬮棿鐨勮窛绂汇€傝繖閲屽彲浠ヤ娇鐢ㄧ浉鍚岀殑鍏紡璁＄畻c鐨勫鏁般€傚湪鎴戜滑鐨勯棶棰樹腑锛岀┖闂翠腑鐨勭鏁ｇ偣鐩搁殧涓€澶╋紝鍥犳h=1銆傚缓璁娇鐢╧<0.1鐨勫€间互淇濇寔鏈夐檺宸垎鍏紡鐨勬煇浜涘噯纭€с€傚洜涓哄鏋渒鐨勫€煎お澶э紝鏈夐檺宸垎鏍煎紡灏变細鍙樺緱涓嶇ǔ瀹氥€傛垜浠幇鍦ㄩ渶瑕佸皢杩欎簺杩戜技鍊兼斁鍏ュ亸寰垎鏂圭▼涓€︹€﹁繖浼氫娇鍏紡鐪嬭捣鏉ユ洿澶嶆潅馃槩杩欐槸鎴戠殑璁＄畻浠ｆ暟杞欢鐨勭粨鏋滐細杩欐槸Perona-MalikPDE鐨勭鏁ｅ舰寮忥紝瓒婃潵瓒婂鏉傘€傛湁娌℃湁鏇村ソ鐨勫姙娉曪紵鎴戜滑鍙互鍋锋噿锛屼娇鐢ㄥ井鍒嗙煩闃点€傚洜涓烘椂闂村簭鍒楁槸涓€缁勭鏁ｇ偣锛屾墍浠ュ彲浠ヤ娇鐢ㄧ煩闃靛悜閲忕Н瀵瑰叾杩涜寰垎銆傚鏋滀綘浠ュ墠浠庢湭鍚杩囪繖涓紝杩欓噷鏈変竴浜涗唬鐮佸睍绀轰簡濡備綍浣跨敤鐭╅樀鍚戦噺涔樼Н璁＄畻澶氶」寮忕殑绠€鍗曞鏁帮細browser'if__name__=='__main__':n=100#绂绘暎鐐逛釜鏁癮=-4#鍖洪棿璧风偣b=4#鍖洪棿缁堢偣#绂绘暎鐐瑰拰闂磋窛x=np.linspace(a,b,n)h=(b-a)/n#寰垎鍑芥暟锛屽叾瑙ｆ瀽瀵兼暟鐢ㄤ簬鏍￠獙f=10*x**3+x**2fx=30*x**2+2*x#褰㈡垚寰垎鐭╅樀Dx=(np.diag(np.ones(n-1),1)-np.diag(np.ones(n-1),-1))/(2*h)#璁＄畻涓婄殑鏁板€煎鏁板彧鏈夊唴閮║x=Dx[1:-1,:]@f#缁樺埗鍥惧舰fig=go.鍥撅紙锛夊浘銆俛dd_trace(go.Scatter(x=x,y=f,name='Function'))fig.add_trace(go.Scatter(x=x,y=fx,name='AnalyitcalDerivative'))fig.add_trace(go.Scatter(x=x[1:-1],y=Ux,name='FiniteDifferenceDerivative',line={'dash':'dash'}))fig.update_layout(yaxis={'鏍囬':'f(x),f_x(x)'},xaxis={'title':'x'},title='FiniteDifferenceDerivativeTest',legend={'x':0,'y':1.075,'orientation':'h'},width=700,height=700,)fig.show()涓婇潰鐨勪唬鐮佸簲璇ヤ骇鐢熶互涓嬭緭鍑猴細涓€闃跺井鍒嗚繎浼煎凡浣跨敤浠ヤ笅浠ｇ爜杞崲涓虹煩闃靛悜閲忎箻绉疍x=(np.diag(np.ones(n-1),1)#u_{r+1,s}terms-np.diag(np.ones(n-1),-1)#u_{r-1,s}terms)/(2*h)#鍙绠楀唴閮ㄧ殑鏁板€煎鏁癠x=Dx[1:-1,:]@f閫氳繃浣跨敤杩欎釜鏈夐檺宸垎鍏紡锛屾垜浠彧鑳藉湪鍐呴儴鐐规壘鍒板鏁般€傚氨鍍忓湪鍩熺殑绗竴涓偣(x=r=0)鏈夎繎浼煎€间竴鏍凤細铏界劧杩欐病鏈夋剰涔夛紝鍥犱负鎵€闇€鐨勮绠楃偣鍦ㄥ煙涔嬪銆備絾杩欏鎴戜滑鏉ヨ涓嶆槸闂鈥斺€斿洜涓烘垜浠彧姹傝В鍐呴儴鐐圭殑鍋忓井鍒嗘柟绋嬶紝鑰岃繖浜涜В鍦ㄧ鐐瑰鏄浐瀹氱殑銆傛垜浠娇鐢ㄤ竴涓畝鍗曠殑灏忕鏁ｆ柟绋嬬粍锛堟瘮濡傛湁5涓鏁ｇ偣锛夛紝涓婇潰鐨勮В閲婂彲鑳芥洿娓呮銆傛渶鍚庝竴涓棶棰橈紝鍗风Н鏄浣曡繘琛岀殑锛熸渶绠€鍗曠殑鏂规硶鏄娇鐢╯cipy.nimage.gaussian_filter锛屼笉杩囨垜杩欓噷閫夋嫨鐨勬柟娉曟槸姹傝В鐑柟绋嬨€傜◢寰湁鐐规暟瀛﹀姛搴曪紝灏卞彲浠ヨ瘉鏄庨珮鏂嵎绉彲浠ユ眰瑙ｇ儹鏂圭▼锛坔ttps://dsp.stackexchange.com...锛夈€傛崲鍙ヨ瘽璇达紝鎴戜滑瑕佽В鍐宠繖涓棶棰樺彲浠ョ敤绂绘暎褰㈠紡琛ㄧず锛屽洜涓洪珮鏂护娉腑鐨勬爣鍑嗗樊锛埾冿級涓庢垜浠眰瑙ｄ笂杩版柟绋嬫墍闇€鐨勨€滄椂闂粹€濋噺鏈夊叧锛屽嵆蟽虏锛埾勶級=2蟿锛屾墍浠ワ紝姹傝В鏃堕棿瓒婇暱锛屾爣鍑嗗樊瓒婂ぇ锛屾椂闂村簭鍒楄秺骞虫粦銆備负浠€涔堣浠ヨ繖绉嶆柟寮忚繘琛屽嵎绉紵PDE鍒板嵎绉殑杩炴帴闈炲父绠€娲侊紒骞朵笖鐢变簬PDE姹傝В閫昏緫鍙互纭紪鐮佷负寰幆锛屽皢鍏跺寘瑁呭湪@numba.jit瑁呴グ鍣ㄤ腑鍙互鎻愰珮璁＄畻鏁堢巼銆侾ython瀹炵幇鐜板湪鎴戜滑宸茬粡瀹屾垚浜嗘暟瀛︽柟闈㈢殑鑹拌嫤宸ヤ綔锛岀紪鐮佸彉寰楅潪甯哥畝鍗曪紒鍗风Н鐨勫疄鐜板涓嬶細importnumpyasnpdefconvolve_PDE(U:np.array,sigma:float=1,k:float=0.05)->np.array:'''閫氳繃姹傝В鐑柟绋嬭繘琛岄珮鏂嵎绉叿鏈塏eumann杈圭晫鏉′欢鍙傛暟----------U:np.array瑕佸鍏舵墽琛屽嵎绉殑鏁扮粍銆俿igma:float,optional楂樻柉鍗风Н鐨勬爣鍑嗗樊銆俴:float,optional鏈夐檺宸垎鏂规鐨勬闀匡紙淇濇寔<0.1浠ヤ繚璇佸噯纭€э級杩斿洖------U:np.array鍗风Н鍑芥暟'''t=0t_end=sigma**2/2whilet
